JHGFRTY

What is the primary condition for two polygons to be considered similar?

Which of the following statements accurately describes the properties of ratios?

In triangle $\triangle ABC$, a line $\overline{DE}$ is drawn parallel to $\overline{BC}$ such that D is on $\overline{AB}$ and E is on $\overline{AC}$. According to the Basic Proportionality Theorem, which of the following relationships is true?

If the sides of two triangles are in proportion, what can be inferred about the triangles?

Given that $\frac{a}{b} = \frac{c}{d}$, which of the following is NOT a valid property of proportions?

Which theorem states that the line joining the midpoints of two sides of a triangle is parallel to the third side and equal to half its length?

If two triangles, $\triangle ABC$ and $\triangle DEF$, are equiangular, what can be concluded?

In a right-angled triangle, if the two sides adjacent to the right angle are 3 cm and 4 cm, what is the length of the hypotenuse?

What is the area of a triangle with a base of 10 cm and a height of 5 cm?

A line is drawn from the midpoint of one side of a triangle, parallel to another side. According to the converse of the Mid-point Theorem, what does this line do to the third side?

Two similar rectangles have lengths in the ratio of 3:4. If the area of the smaller rectangle is 18 cm$^2$, what is the area of the larger rectangle?

Which of the following is a necessary condition for two triangles to be congruent?

If the diagonals of a rhombus are 6 cm and 8 cm, what is its area?

Given a trapezium with parallel sides of lengths 10 cm and 12 cm, and a height of 4 cm, find its area.

In $\triangle ABC$, point D lies on AB and point E lies on AC such that $\overline{DE} \parallel \overline{BC}$. If $AD = x$, $DB = x + 2$, $AE = 3$, and $EC = 5$, find the value of $x$.

What is the relationship between the areas of two triangles with equal heights?

Given $\triangle ABC \sim \triangle DEF$, with $AB = 6$, $BC = 8$, $DE = 9$. Find the length of $EF$.

What is the area of a square with a side length of 7 cm?

If $\frac{x}{4} = \frac{9}{6}$, what is the value of $x$?

If the corresponding sides of two triangles are proportional, what ensures their similarity?

In a right triangle $ABC$ with right angle at $A$, $AB = 5$ and $AC = 12$. Find the length of $BC$.

Which of the following polygons does NOT have diagonals that bisect each other?

If the area of a parallelogram is 48 cm$^2$ and its base is 8 cm, what is its height?

In (\triangle ABC), DE is parallel to BC, with D on AB and E on AC. If AD:DB = 2:3 and AE = 4 cm, find EC.

The sides of a triangle are 5, 12, and 13. Is the triangle a right triangle?

If $\triangle ABC$ and $\triangle PQR$ are similar such that $AB/PQ = BC/QR = CA/RP = k$, what does $k$ represent?

What is the converse of the Pythagorean Theorem used to determine?

Consider a quadrilateral where diagonals intersect at right angles. Which of the following is always true for such quadrilaterals?

A pole casts a shadow of 15 meters. If a 2-meter stick casts a shadow of 1 meter, how tall is the pole?

If two triangles have the same base and are equal in area, what can be said about their vertices opposite the base?

In a quadrilateral ABCD, if $\angle A = \angle B = \angle C = 90^{\circ}$, what type of quadrilateral is it?

Suppose triangles $\triangle ABC$ and $\triangle A'B'C'$ satisfy $AB = k \cdot A'B'$, $BC = k \cdot B'C'$, and $CA = k \cdot C'A'$ for some constant $k > 0$. Which of the following statements must be true?

Given a triangle ABC, D is a point on AB and E on AC such that DE is parallel to BC. If AD = 2x, DB = x+1, AE = x and EC = x-1, what is x?

Let $\triangle ABC$ be a triangle with sides $AB = 13$, $BC = 14$, and $CA = 15$. Let $D$ be the foot of the altitude from $A$ to $BC$. Find the length of $BD$.

Point $P$ is inside rectangle $ABCD$. If $PA = 3$, $PB = 4$, and $PC = 5$, what is the value of $PD$?

In $\triangle ABC$, $AB = 5$, $AC = 7$, and $BC = 8$. If $M$ is the midpoint of $BC$, find the length of $AM$.

In geometric terms, what fundamentally defines 'proportion'?

Consider $\triangle ABC$ where point $D$ lies on side $AB$ and point $E$ lies on side $AC$. If $DE$ is parallel to $BC$, and $AD = 4$, $DB = 6$, and $AE = 5$, what is the length of $EC$?

Two polygons are similar. Which statement regarding their corresponding angles and sides must be true?

In $\triangle ABC$, point $D$ is on $AB$ and $E$ is on $AC$, with $DE \parallel BC$. If $AD:DB = 2:5$ and $AE:EC = 3:x$, find the value of $x$ that ensures $DE$ is parallel to $BC$ according to the Basic Proportionality Theorem.

Consider $\triangle ABC$ with median $AD$ to side $BC$. Point $E$ lies on $AD$ such that $AE:ED = 2:1$. If line $BE$ is extended to intersect $AC$ at point $F$, determine the ratio $AF:FC$ using Menelaus' theorem or otherwise.

What is the simplified form of a ratio comparing 25 cm to 1 meter?

Which property of proportions is demonstrated by the equation if $\frac{a}{b} = \frac{c}{d}$, then $\frac{d}{c} = \frac{b}{a}$?

In $\triangle ABC$, a line segment $DE$ is parallel to side $BC$, with $D$ on $AB$ and $E$ on $AC$. If $AD = 4$, $DB = 6$, and $AE = 5$, what is the length of $EC$?

Two rectangles are similar. The first rectangle has a length of 15 cm and a width of 10 cm. The second rectangle has a length of 18 cm. What is the width of the second rectangle?

The area of a triangle is given by which of the following formulas, where 'b' is the base and 'h' is the height?

What is the area of a parallelogram with a base of 12 cm and a height of 8 cm?

If the diagonals of a rhombus are 10 cm and 14 cm, what is the area of the rhombus?

A square has a side length of 9 cm. What is the area of the square?

Which of the following transformations always preserves similarity?

In $\triangle ABC$, $D$ is the midpoint of $AB$ and $E$ is the midpoint of $AC$. If $BC = 10$ cm, what is the length of $DE$?

What is the area of a trapezium with parallel sides of lengths 8 cm and 14 cm, and a height of 6 cm?

Which of the following theorems is used to prove that equiangular triangles are similar?

What does the Converse of the Mid-point Theorem state?

How does the area of two triangles with the same base relate if they are between the same parallel lines?

Given $\triangle ABC \sim \triangle DEF$, with $AB = 8$, $BC = 12$, and $DE = 10$. Find the length of $EF$.

According to the properties of proportions, if $\frac{a}{b} = \frac{c}{d}$, which of the following is true?

If a line divides two sides of a triangle proportionally, then:

Given that two triangles, $\triangle ABC$ and $\triangle DEF$, have proportional sides, what can be definitively stated?

In right triangle $\triangle ABC$, where $\angle B = 90^{\circ}$, if $AB = 8$ cm and $BC = 6$ cm, find the length of $AC$.

How does the area of a triangle change if its height is doubled while its base remains constant?

Consider $\triangle ABC$ with $AB = 9$, $AC = 12$. Point $D$ lies on $AB$ such that $AD = 6$. If $DE$ is parallel to $BC$, what is the length of $AE$?

Which of the following statements is true regarding similar triangles?

In geometry, what is a 'ratio' fundamentally used to express?

What significant condition must be validated to establish two triangles as 'similar'?

What characteristic defines when two ratios are considered to be in 'proportion'?

If a line is drawn parallel to one side of a triangle intersecting the other two sides, what does the Basic Proportionality Theorem determine about the division of these sides?

What does the property of 'Inverted Proportion' imply given a proportion $\frac{a}{b} = \frac{c}{d}$?

How is 'height' defined in the context of triangles concerning area calculation?

Regarding triangles with equal areas, what is a definitive relationship between their bases and heights?

In $\triangle ABC$, point $D$ is on $AB$ and $E$ is on $AC$ such that $DE$ is parallel to $BC$. If $AD = 2x$, $DB = x+3$, $AE = x$ and $EC = x+2$, what is the value of $x$?

Given $\triangle ABC$ with a point $D$ on $AB$ and $E$ on $AC$ such that $DE \parallel BC$. If $AD = 5$, $DB = 3$, and $AE = 7$, find the length of $EC$.

The sides of a triangle are 7, 24, and 25. Is this a right triangle? Why or why not?

Two poles of heights 6 meters and 11 meters stand vertically on a plane ground. If the distance between their feet is 12 meters, what is the distance between their tops?

If the area of a square is 64 cm$^2$, what is the length of its diagonal?

Consider a quadrilateral $ABCD$ where the diagonals $AC$ and $BD$ intersect at point $E$. Which condition would definitively prove that $ABCD$ is a parallelogram?

Let $\triangle ABC$ be a triangle. Point $D$ lies on $AB$ and point $E$ lies on $AC$ such that $DE \parallel BC$. If $AD = x + 3$, $DB = 3x + 1$, $AE = x$ and $EC = 2x - 2$, find the value of $x$ that satisfies these conditions.

Consider triangle ABC, with vertices A(0, 0), B(6, 0), and C(0, 8). A line is drawn parallel to BC that bisects the area of triangle ABC. What is the y-intercept of this line?

In triangle $\triangle ABC$, $AB = AC$. Point $D$ is on $AC$ such that $BD$ bisects angle $ABC$. If $\angle BDC = 72^{\circ}$, what is $\angle A$?

What distinguishes a ratio from other mathematical comparisons?

In the context of proportions, what does the property of 'Alternating Proportion' allow?

Consider a quadrilateral $ABCD$. Which condition is sufficient to prove it is a kite?

In $\triangle ABC$, point $D$ lies on side $AB$ and point $E$ lies on side $AC$. $DE$ is parallel to $BC$. Given $AD = x + 2$, $DB = 3x - 2$, $AE = x$, and $EC = 2x - 1$, find the value of $x$.

Let $ABCD$ be a square, and let $P$ be a point inside the square such that $PA = 1$, $PB = 2$, and $PC = 3$. What is the length of the side of the square?

What is a ratio?

Which of the following is NOT a fundamental property of ratios?

What does it mean for two ratios to be in proportion?

Given the proportion $\frac{a}{b} = \frac{c}{d}$, which property is demonstrated by the transformation to $ad = bc$?

If $\frac{p}{q} = \frac{r}{s}$, which of the following correctly applies the property of Alternating Proportion?

In geometry, what does the Basic Proportionality Theorem state regarding a line drawn parallel to one side of a triangle?

What are the steps to solving proportional problems in the correct order?

In triangle $\triangle ABC$, let $D$ be a point on $AB$ and $E$ be a point on $AC$ such that $DE \parallel BC$. If $AD = 2$, $DB = 3$, and $AE = 4$, find the length of $EC$.

Which statement accurately describes a polygon?

Which formula correctly calculates the area of a parallelogram, where $b$ is the base and $h$ is the height?

What is the area of a rhombus with diagonals of length 8 cm and 10 cm?

If a square has a side length of 11 cm, what is its area?

The parallel sides of a trapezium are 6 cm and 8 cm, and the height is 5 cm. What is the area of the trapezium?

In a kite, the lengths of the diagonals are 12 cm and 16 cm. What is the area of the kite?

For polygons to be considered similar, what conditions must be met?

Triangle $\triangle ABC$ has a height of 5 cm and a base $BC$ of 8 cm. Another triangle $\triangle DEF$ has the same height but a base $EF$ of 12 cm. What is the ratio of the area of $\triangle ABC$ to the area of $\triangle DEF$?

What can be said about triangles that have equal bases and lie between the same parallel lines?

What does the Mid-point Theorem state about the line joining the midpoints of two sides of a triangle?

Given a triangle $\triangle ABC$, where $D$ and $E$ are points on sides $AB$ and $AC$ respectively, and $DE \parallel BC$. If $AD = 3$, $DB = 5$, and $AE = 4$, what is the length of $EC$?

In $\triangle ABC$, $AB = 4$ cm and $AC = 6$ cm. A line $DE$ is drawn parallel to $BC$, intersecting $AB$ at $D$ and $AC$ at $E$. If $AD = 1.5$ cm, what is the length of $AE$?

Two triangles, $\triangle ABC$ and $\triangle DEF$, are given with $\angle A = \angle D$, $\angle B = \angle E$, and $\angle C = \angle F$. What theorem allows us to conclude that the triangles are similar?

If the sides of two triangles are in proportion, which of the following statements must be true?

Which of the following statements correctly applies the Converse of the Mid-point Theorem?

Consider $\triangle ABC$ and $\triangle DEF$. If $AB = 6$, $BC = 8$, $AC = 10$, and $DE = 9$, $EF = 12$, and $DF = 15$, are the triangles similar, and if so, by what criterion?

In a right-angled triangle, the two sides adjacent to the right angle (legs) measure 5 cm and 12 cm. According to the Pythagorean Theorem, what is the length of the hypotenuse?

Which of the following is required to prove two triangles are similar?

Given $\triangle ABC$ and $\triangle DEF$ are similar, and the ratio of their corresponding sides is $2:3$. If the area of $\triangle ABC$ is 20 cm$^2$, what is the area of $\triangle DEF$?

Consider two triangles, $\triangle ABC$ and $\triangle XYZ$, with the following side lengths: $AB = 4$, $BC = 6$, $CA = 8$ and $XY = 6$, $YZ = 9$, $ZX = 12$. Determine if $\triangle ABC \sim \triangle XYZ$ and identify the similarity criterion that applies.

Given a right triangle with legs of lengths $a$ and $b$, and hypotenuse of length $c$, what does the Pythagorean Theorem state?

In a right triangle $\triangle ABC$ with $\angle A = 90^{\circ}$, $AB = 8$ and $AC = 15$. What is the length of the hypotenuse $BC$?

What does the Converse of the Pythagorean Theorem allow us to determine?

If a triangle has sides of length 9, 12, and 15, is it a right triangle? If so, which side is the hypotenuse?

What is the core idea behind the proof of the Pythagorean Theorem presented?

Consider $\triangle ABC$ where $BC = a$, $AC = b$, and $AB = c$. Given that $a^4 + b^4 = c^4$, what can be inferred about $\triangle ABC$?

In triangle $\triangle ABC$, point $D$ lies on $AB$ and $E$ lies on $AC$, with $DE \parallel BC$. If $AD = x$, $DB = x + 3$, $AE = x - 2$ and $EC = x + 1$, what is the value of $x$?

Suppose in a triangle $ABC$, point $D$ is on $AB$ and $E$ is on $AC$ such that $DE$ is parallel to $BC$. Given $AD = 4$, $DB = x - 3$, $AE = 8$, and $EC = x + 1$, find the value of $x$.

Suppose in $\triangle ABC$, $D$ is a point on $AB$ and $E$ is a point on $AC$ such that $DE \parallel BC$. If $AD = x+1$, $DB = 3$, $AE = x+3$, and $EC = 5$, what is the value of $x$?

Assume you have a triangle $ABC$, where a line $DE$ runs parallel to $BC$, intersecting $AB$ at $D$ and $AC$ at $E$. If $AD = 2x + 1$, $DB = 3x - 1$, $AE = 2$, and $EC = 3$, find the value of $x$.

In similar polygons, what is the relationship between their corresponding sides?

According to the properties of proportion, if $\frac{a}{b} = \frac{c}{d}$, which of the following is a valid transformation demonstrating 'Alternating Proportion'?

In the context of the Pythagorean Theorem, which of the following statements is fundamentally true about right-angled triangles?

Given triangle $\triangle ABC$ with point $D$ on side $AB$ and point $E$ on side $AC$ such that $DE \parallel BC$. If $AD = x + 2$, $DB = 3$, $AE = x + 4$, and $EC = 5$, find the value of $x$.

Consider a scalene triangle $\triangle ABC$. Points $D$, $E$, and $F$ lie on sides $BC$, $CA$, and $AB$ respectively, such that $AD$, $BE$, and $CF$ are concurrent. If $\frac{AF}{FB} = \frac{2}{3}$ and $\frac{BD}{DC} = \frac{4}{5}$, what is the ratio of $\frac{CE}{EA}$?