Mathematics Problem Solving
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Questions and Answers

Determine the zeroes of the quadratic polynomial after analyzing its intersection points: (-6, 0), (0, -30), (4, -20), and (6, 0).

-6 and 6

For the system of equations given by $3x - ky - 7 = 0$ and $6x + 10y - 3 = 0$, what value of $k$ results in inconsistency?

-10

Explain why the statement 'From a point inside a circle, only two tangents can be drawn' is true or false.

False, only one tangent can be drawn from a point inside a circle.

If the nth term of an arithmetic progression (AP) is given as $7n - 4$, calculate the common difference.

<p>7</p> Signup and view all the answers

If a right circular cone's radius is 5 cm and its volume equals that of a sphere with the same radius, calculate the height of the cone.

<p>20 cm</p> Signup and view all the answers

In a circle, if PT is a tangent and ∠TPO = 35°, what is the value of ∠x?

<p>55°</p> Signup and view all the answers

What shape is formed if the diagonals of a quadrilateral divide each other proportionally?

<p>A parallelogram.</p> Signup and view all the answers

In triangle ABC, with DE parallel to BC and given AD = 2 cm, BD = 3 cm, and BC = 7.5 cm, find the length of DE.

<p>5 cm</p> Signup and view all the answers

If the HCF of 2520 and 6600 is 40, and the LCM is 252k, what is the value of k?

<p>66</p> Signup and view all the answers

What is the probability of randomly choosing an odd prime number from the digits 1 to 9?

<p>1/3</p> Signup and view all the answers

How can you determine the length of the segment connecting two intersection points of chords in a circle?

<p>You can use the power of a point theorem, which states that the product of the lengths of the segments of one chord equals the product of the lengths of the segments of the other chord.</p> Signup and view all the answers

What is the quadratic formula used to find the roots of a quadratic equation?

<p>The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are coefficients of the quadratic equation $ax^2 + bx + c = 0$.</p> Signup and view all the answers

How do you find the median of a data set that is organized in a frequency distribution?

<p>You need to identify the cumulative frequency to locate the median class and then use interpolation if the median falls within that class.</p> Signup and view all the answers

In the context of probability, how would you determine the chances of drawing an Ace from a standard deck of playing cards?

<p>The probability is calculated by dividing the number of favorable outcomes (4 Aces) by the total number of outcomes (52 cards), resulting in $P(Ace) = \frac{4}{52} = \frac{1}{13}$.</p> Signup and view all the answers

What relationship exists between the angles formed by a tangent and a chord at their intersection point on a circle?

<p>The angle formed by the tangent and the chord is equal to the angle in the alternate segment of the circle.</p> Signup and view all the answers

What is the HCF of 480 and 720 using the prime factorization method?

<p>The HCF of 480 and 720 is 240.</p> Signup and view all the answers

What value of m satisfies the equation HCF(85, 238) = 85m - 238?

<p>The value of m is 3.</p> Signup and view all the answers

How would you calculate the perimeter of a sector if the radius is halved and the angle is doubled?

<p>The perimeter remains the same due to the proportional changes in radius and angle.</p> Signup and view all the answers

Find the point(s) on the X-axis at a distance of $ rac{ ext{approximately}}{ ext{sqrt{41}}}$ units from the point (8, -5).

<p>The points are (8 + 6.4, 0) and (8 - 6.4, 0), which are approximately (14.4, 0) and (1.6, 0).</p> Signup and view all the answers

Demonstrate that points A(-5, 6), B(3, 0), and C(9, 8) form an isosceles triangle.

<p>AB = AC = 10 units.</p> Signup and view all the answers

What formula expresses the area of the triangle with vertices at A(-5, 6), B(3, 0), and C(9, 8)?

<p>Area = $0.5 * |(-5)(0-8) + (3)(8-6) + (9)(6-0)|$.</p> Signup and view all the answers

What is the result of evaluating $2 imes ext{sin}(60^ ext{o}) imes ext{tan}^2(30^ ext{o}) / ext{sec}(45^ ext{o})$?

<p>The result is 2.</p> Signup and view all the answers

How can you form a quadratic polynomial whose zeroes are $1/α$ and $1/β$ from the polynomial $6x^2 - 5x + 1$?

<p>The new polynomial is $x^2 - rac{5}{6}x + rac{1}{6}$.</p> Signup and view all the answers

If cosθ + sinθ = 1, what does this imply about the values of cosθ and sinθ?

<p>cosθ = 0 and sinθ = 1.</p> Signup and view all the answers

Calculate the area of the face of a clock described by a minute hand of length 18 cm in 35 minutes.

<p>The area is 660 cm².</p> Signup and view all the answers

Show that √3 is an irrational number.

<p>Assume √3 = p/q leads to a contradiction since p and q must be integers.</p> Signup and view all the answers

Find the speeds of two cars if one is 180 km from the other and they meet in 1 hour when traveling towards each other.

<p>Speeds are 20 km/h and 10 km/h.</p> Signup and view all the answers

What do the lengths of tangents drawn from an external point to a circle being equal imply geometrically?

<p>They show that the tangents are congruent.</p> Signup and view all the answers

If a boy sees a balloon from a height of 1.35 m, how can one find the height of the balloon given the angle of elevation?

<p>By applying trigonometry to the angle of elevation.</p> Signup and view all the answers

For the given frequency distribution of monthly expenditures, how do you compute the mean?

<p>Use the formula: Mean = Σ(fx) / Σf, where f is frequency and x is the mid-point.</p> Signup and view all the answers

Determine if it is feasible to arrange jars in an arithmetic progression if the number pattern is consistent.

<p>Yes, if the common difference is a positive integer.</p> Signup and view all the answers

What can be concluded about the solution set of the linear equations $x + 2y + 5 = 0$ and $-3x + 6y - 1 = 0$?

<p>These equations have infinitely many solutions because they are dependent.</p> Signup and view all the answers

Calculate the common difference of the arithmetic progression given by $ rac{1}{2x}, rac{1 - 4x}{2x}, rac{1 - 8x}{2x},...$.

<p>The common difference is $-2$.</p> Signup and view all the answers

What is the probability that two dice thrown together show different numbers?

<p>The probability is $ rac{5}{6}$.</p> Signup and view all the answers

If the probability of guessing the correct answer to a question is $ rac{1}{3}$, what is the probability of guessing incorrectly?

<p>The probability of guessing incorrectly is $ rac{2}{3}$.</p> Signup and view all the answers

Determine the value of $x$ if the probability of not guessing the correct answer to a question, given as $ rac{1}{3}$, implies certain conditions.

<p>The value of $x$ is 2.</p> Signup and view all the answers

What is the arithmetic progression (AP) representing the number of jars in each layer, and what is the common difference?

<p>The AP is 3, 6, 9, ..., with a common difference of 3.</p> Signup and view all the answers

Is it possible to arrange 34 jars in a layer? Justify your answer.

<p>No, 34 is not achievable as it does not fit the arithmetic progression of layers with a common difference of 3.</p> Signup and view all the answers

Find the expression for the total number of jars in a layer if there are n rows.

<p>The expression for total jars is $3n$, hence $S_n = 3 + 6 + 9 + ... + 3n = \frac{3n(n+1)}{2}$.</p> Signup and view all the answers

How many jars are present in the 5th layer if 3 more jars are added to each layer?

<p>In the 5th layer, there will be 18 jars (15 original + 3 additional).</p> Signup and view all the answers

Given the data on expenditures, what is the range of expenditure for the unfilled bracket?

<p>The expenditure range for the unfilled bracket is $2500 - 3000$.</p> Signup and view all the answers

How can you prove that $ riangle ADPQ ext{ is similar to } riangle ADEF$?

<p>By showing that corresponding angles of the triangles are equal, due to the parallel lines.</p> Signup and view all the answers

Calculate the slant height of the conical part if the radius is 1.5 m and height is 7 m.

<p>The slant height is approximately 7.5 m, calculated using $l = \sqrt{r^2 + h^2} = \sqrt{1.5^2 + 7^2}$.</p> Signup and view all the answers

What is the total capacity of the silo to store grains, combining both parts?

<p>The total capacity is approximately $81.3 m^3$.</p> Signup and view all the answers

What are the coordinates of the point that divides the line segment joining A(2, 3) and B(5, 7) internally in the ratio 1:2?

<p>(3, 5)</p> Signup and view all the answers

Calculate the probability of drawing the queen of hearts from a standard deck of cards.

<p>$\frac{1}{52}$</p> Signup and view all the answers

For the linear equations 2x + y = 13 and 4x - y = 17, what are the values of x and y?

<p>x = 5, y = 3</p> Signup and view all the answers

What relation exists for point P(x, y) to be equidistant from points A(7, 1) and B(3, 5)?

<p>(x - 7)² + (y - 1)² = (x - 3)² + (y - 5)²</p> Signup and view all the answers

Given that the sum of the first 7 terms of an AP is 49, can you find the sum of its first 20 terms?

<p>Sum = 140</p> Signup and view all the answers

What are the zeroes of the polynomial x² - 15?

<p>$\pm \sqrt{15}$</p> Signup and view all the answers

How would you graphically solve the system of equations x - y + 1 = 0 and x + y = 5?

<p>Plot the lines and find their intersection point (2, 3).</p> Signup and view all the answers

If the ratio of the 10th term to the 30th term of an AP is 1:3, what does that tell you about the common difference?

<p>Common difference d = (3a / 20)</p> Signup and view all the answers

What is the ratio in which the line segment joining the points (5, 3) and (-1, 6) is divided by the y-axis?

<p>The ratio is 3:2.</p> Signup and view all the answers

Find the coordinates of point R on line segment PQ where PR = 2QR, given points P(-2, 5) and Q(3, 2).

<p>The coordinates of R are (1, 3).</p> Signup and view all the answers

Prove the identity $ an^{2} heta = an^{2} heta$ using the expression $ an^{2} heta = rac{ ext{sin}^{2} heta}{ ext{cos}^{2} heta}$.

<p>By using the identity $ ext{sin}^{2} heta + ext{cos}^{2} heta = 1$, the proof confirms the equation.</p> Signup and view all the answers

Why do tangents drawn at the endpoints of a chord of a circle make equal angles with that chord?

<p>This occurs due to the properties of tangents and the inscribed angle theorem.</p> Signup and view all the answers

What is the original duration of a flight that experienced a 100 km/h speed reduction resulting in a 30 min increase in flight time?

<p>The original duration of the flight was 6 hours.</p> Signup and view all the answers

Find the fraction if the denominator is one more than twice the numerator and the sum of the fraction and its reciprocal is $ rac{16}{3}$.

<p>The fraction is $ rac{4}{5}$.</p> Signup and view all the answers

Calculate the height of the tower if the angle of elevation to its top is 60° and to its base is 45°, with the building height being 20 m.

<p>The height of the tower is 20 m.</p> Signup and view all the answers

What is the surface area of a medicine capsule which is a cylinder with hemispherical ends, given the total length is 14 mm and diameter is 4 mm?

<p>The surface area is $98 ext{ mm}^{2}$.</p> Signup and view all the answers

What is the value of $a$ in the expression $H = 2 imes 3 imes 5^a imes 7^b$ if $LCM(H, C) = 3780$?

<p>1</p> Signup and view all the answers

List the zeroes of the quadratic polynomial $2x^2 - 3x - 9$.

<p>$ rac{3 ext{ } ext{±} ext{ } ext{sqrt}(93)}{4}$</p> Signup and view all the answers

Calculate the height of a tower if the angle of elevation from a point 30m away is 60°.

<p>$30 ext{ } ext{sqrt}(3)$</p> Signup and view all the answers

If $ an( heta + rac{ ext{PI}}{3})$ gives a certain value based on the angles $ heta$ where $ rac{1}{2}$ and $ rac{ ext{sqrt}(3)}{2}$, compute this value.

<p>$ ext{not defined}$</p> Signup and view all the answers

What is the length of segment DE in triangle ABC if $ rac{AD}{AB} = rac{DE}{BC}$?

<p>2.5</p> Signup and view all the answers

Determine the value of $k$ if $HCF(2520, 6600) = 40$ and $LCM(2520, 6600) = 252k$.

<p>165</p> Signup and view all the answers

Find a pair of irrational numbers that when multiplied yield a rational number.

<p>($ rac{ ext{sqrt}(3)}{3}, rac{ ext{sqrt}(3)}{ ext{sqrt}(27)})$</p> Signup and view all the answers

Calculate the probability of selecting an odd prime number from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.

<p>$ rac{2}{9}$</p> Signup and view all the answers

Flashcards

Tangent to a Circle

A line that touches a circle at exactly one point.

Proportional Diagonals

In a quadrilateral, the diagonals divide each other in the same proportion.

Similar Triangles

Triangles with the same shape but different sizes; parallel lines create similar triangles.

Finding DE

If DE is parallel to BC in triangle ABC, then the length of DE is proportional to the lengths of AD, BD, and BC.

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Probability of Odd Prime

The likelihood of selecting an odd prime number from a given set of digits.

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Zeroes of a quadratic polynomial

The values of x where the polynomial equals zero (when graphed, these are the x-intercepts).

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Inconsistent system of equations

A system of equations with no solution.

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Common difference (AP)

The constant difference between consecutive terms in an arithmetic progression (AP).

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Volume of a cone

One-third the product of the area of the base and the height. V = (1/3)πr²h where r is the radius and h is the height

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Intersecting Chords Theorem

When two chords intersect inside a circle, the product of the segments of one chord is equal to the product of the segments of the other chord.

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Quadratic Polynomial Zeroes

The values of x where a quadratic polynomial equals zero, also known as the x-intercepts on the graph.

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Frequency Distribution Median

The midpoint of a data set arranged in order of frequency. If there are an even number of data points, the median is the average of the two middle values.

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Probability of an Event

The likelihood of an event occurring, expressed as a fraction or decimal between 0 and 1.

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HCF of consecutive even numbers

The highest common factor (HCF) of two consecutive even natural numbers is always 2. This is because even numbers are divisible by 2, and consecutive even numbers share the common factor 2.

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Sector Perimeter Remains the Same

If the radius of a sector is halved and the central angle is doubled, the perimeter of the sector remains unchanged. This is due to the relationship between the arc length and the radius, and the fact that the radius is halved, and angle is doubled for a sector area of $\pi r^2$, $\pi r^2$ or $ rac{1}{2} \pi r^2$, which cancels out the change in length to area ratio.

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Prime Factorization Method

A method of finding the HCF and LCM of numbers by breaking them down into their prime factors. The HCF is the product of the common prime factors raised to their lowest powers, and the LCM is the product of all prime factors raised to their highest powers.

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Expressing HCF as a Linear Combination

The HCF of two numbers can always be expressed as a linear combination of the two numbers. This means that the HCF can be written in the form ax + by, where a and b are integers, and x and y are the two numbers.

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Probability of an Odd Product

When two dice are rolled, the probability of getting an odd product is determined by the probability of getting an odd number on both dice. An odd product is only possible if both dice land on odd numbers.

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Counting Three-Digit Integers

To count three-digit integers with a specific hundredths digit and unit digit, fix those digits and vary the tens digit. The probability of selecting one such integer out of all three-digit integers is calculated by dividing the number of favorable outcomes by the total number of possible outcomes.

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Evaluating Trigonometric Expressions

Evaluating trigonometric expressions involves substituting the values of trigonometric functions at specific angles into the expression and performing algebraic operations. Knowing the values of trigonometric functions for common angles is essential.

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Distance Between a Point and the X-axis

The distance between a point and the X-axis is the absolute value of the y-coordinate of that point. This is because the X-axis represents all points with a y-coordinate of 0.

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Trigonometric Identity

An equation that is true for all values of the variables involved. It's a relationship between trigonometric functions.

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Area of a Sector

The area of a portion of a circle bounded by two radii and an arc. It's like a slice of pizza.

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Area of a Segment

The area of a portion of a circle bounded by a chord and an arc. It's like a slice of pizza with a crust.

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Irrational Number

A number that cannot be expressed as a simple fraction (a/b) where a and b are integers. Its decimal representation is non-repeating and non-terminating.

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System of Linear Equations

A set of two or more linear equations with the same variables.

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Speed of Two Objects

The rate at which objects move, often measured in km/h or m/s.

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Mean and Median

Measures of central tendency. Mean is the average, found by summing values and dividing by the number of values. Median is the middle value when data is ordered.

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Linear Equations: Unique Solution

A pair of linear equations has a unique solution if the lines they represent intersect at a single point. This means there's only one value for 'x' and one value for 'y' that satisfies both equations.

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Arithmetic Progression (AP) Common Difference

The common difference in an arithmetic progression (AP) is the constant value added to each term to get the next term. It's the 'step' between consecutive numbers in the sequence.

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Probability: Different Numbers on Dice

When two dice are thrown, the probability of getting different numbers is calculated by considering all possible outcomes where the dice show different values and dividing by the total number of possible outcomes.

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Probability: Guessing Correct Answer

If the probability of guessing the correct answer is 'x', then the probability of not guessing the correct answer is 1 - x. The sum of these probabilities must always equal 1.

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Arithmetic Progression (AP)

A sequence of numbers where the difference between any two consecutive terms is constant. This constant difference is called the common difference.

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Layer's Jars from 'n' Rows

In a pattern where each layer has a fixed number of jars added to the previous layer, the total number of jars in a layer with 'n' rows is given by the formula: n * (first term + (n - 1) * common difference).

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Proportional Sides

In similar triangles, corresponding sides are in the same ratio. This means if one side of a triangle is twice as long as the corresponding side in the other triangle, all other corresponding sides will also be twice as long.

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Median of a Triangle

A line segment drawn from a vertex of a triangle to the midpoint of the opposite side.

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Calculating Slant Height

In a cone, the slant height is the distance from the apex (tip) of the cone to a point on the circle of the base. It can be calculated using the Pythagorean theorem: slant height² = height² + radius²

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Curved Surface Area of a Cone

The area of the cone's curved surface, excluding the base. It's calculated using the formula: π * radius * slant height.

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Internal Division Point

The point that divides a line segment into two parts in a given ratio, where both parts are on the same side of the point.

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Equidistant Points

Points that are the same distance away from a given point or line.

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Diameter of a Circle

A line segment that passes through the center of a circle and has endpoints on the circle.

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Sum of First 'n' Terms of AP

The total value obtained by adding the first 'n' terms of an arithmetic progression (AP).

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Zeroes of a Polynomial

Values of the variable (usually 'x') that make the polynomial equal to zero.

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Graphical Solution of Equations

Finding the solution to a system of equations by plotting the lines represented by the equations on a graph.

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LCM of Numbers

The Least Common Multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers.

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Angle of Elevation?

The angle of elevation is the angle formed between the horizontal line of sight and a line of sight to an object above the horizontal.

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Proportionality

Two quantities are proportional if they have a constant ratio, meaning they change at the same rate.

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HCF and LCM Relationship

The product of two numbers is equal to the product of their HCF and LCM.

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Ratio of Division

The ratio in which a line segment is divided by a point is the ratio of the lengths of the two segments formed.

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Section Formula

Given two points A(x1, y1) and B(x2, y2), the coordinates of a point C that divides the segment AB in the ratio m:n are ((mx2 + nx1)/(m+n), (my2 + ny1)/(m+n)).

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Trigonometric Identity Proof

Prove that an equation involving trigonometric functions is true for all values of the angle.

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Tangents to a Chord

The tangents drawn at the end points of a chord of a circle make equal angles with the chord.

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Speed, Distance, and Time

The relationship between speed, distance, and time is: Distance = Speed x Time; Speed = Distance/Time; Time = Distance/Speed.

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Solving a Quadratic Equation

A quadratic equation is an equation of the form ax² + bx + c = 0, where a, b, and c are constants. The solutions to the equation are called the roots.

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Basic Proportionality Theorem

If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.

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