Geometry and Volume of Shapes Quiz

A parallelepiped with all rectangular faces. It has three dimensions: length (l), breadth (b), and height (h).

The line segment connecting opposite vertices of a cuboid. There are four body diagonals, all equal in length.

The distance along a face of a cuboid, connecting opposite vertices. There are three pairs of face diagonals.

A solid with two parallel, congruent ends, and side faces that are parallelograms. The ends can be triangles, quadrilaterals, or polygons.

A prism where the side edges are perpendicular to the ends. This results in rectangular side faces.

A special case of a cuboid where all faces are squares. It has only one dimension: side length (a).

The line segment connecting opposite vertices of a cube. There are four body diagonals, all equal in length.

The line segment connecting opposite vertices of a cube's face. All face diagonals are equal in length.

The angle bisectors of two angles of a triangle divide the third angle into two angles which are equal to half the sum of the first two angles.

The area of a triangle is directly proportional to the base and height. If the base is increased by a factor of 2.5, the area will also increase by the same factor.

The median of a triangle divides the triangle into two triangles with equal areas. The line segment connecting the midpoint of a side to the opposite vertex is called a median.

In a triangle, the ratio of the lengths of the corresponding sides of similar triangles is equal. If two angles of one triangle are equal to two angles of another triangle, then the two triangles are said to be similar.

The Pythagorean theorem states that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. The hypotenuse is the longest side of a right-angled triangle.

When a triangle is moved along a line, the area of the overlapping region is equal to half the area of the original triangle.

Parallel lines cut by a transversal create corresponding angles. They are equal in measure. The ratio of lengths of corresponding line segments is the same.

The angle bisector theorem states that the bisector of an angle of a triangle divides the opposite side into segments that are proportional to the other two sides of the triangle. The angle bisector is a line segment that divides an angle into two equal angles.

In a triangle, if the sum of the lengths of all the sides is equal to the sum of the squares of the lengths of all the sides, then the triangle is an equilateral triangle.

The perimeter of a triangle is the total length of all its sides. In the given case, we need to find the sum of the lengths of the sides of triangle PQR.

The line segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.

Angles at the same position in two congruent triangles are equal, as well as corresponding sides being equal. This allows us to find missing angles and sides in similar triangles.

A perpendicular line drawn from the vertex of a triangle to its opposite side divides the triangle into two right-angled triangles. This helps us find relationships between angles and sides in a triangle.

The measure of the angle at the centre of a circle is twice the measure of the angle at the circumference when both angles are subtended by the same arc.

The ratio of the length of the line segment AD to the length of the line segment DC is 3:2.

The sum of the angles at the five points of a star formed by connecting five points on a circle is 540 degrees.

The diameter of the inscribed circle of an isosceles trapezoid with parallel sides of lengths 75 and 108 units is 91.5.

The ratio of the area of the circle circumscribing the square to the area of the circle inscribed in the equilateral triangle is 27:8.

The length of line segment LC is 12 cm. Given the cube is 5/8th full with liquid, calculate the height of the liquid and use ratios to find the length of LC.

The radius of the circle is [(x² + y² - xy)/3]1/2 units.

The value of the ratio AD:DC is 7:3.

The length of AB is 26 + 2√38 units.

A line that touches a circle at exactly one point.

The radius drawn to the point of contact of a tangent is perpendicular to the tangent.

If two tangents are drawn to a circle from an external point, the lengths of the tangent segments are equal.

If a secant and a tangent are drawn from an external point to a circle, the product of the lengths of the secant segment and its external segment is equal to the square of the tangent segment.

The angle between a tangent and a chord drawn from the point of contact is equal to the angle subtended by that chord in the alternate segment of the circle.

In a right-angled triangle with sides AB = 7, AC = 25, and AD as the angle bisector of angle A, AD is also a perpendicular bisector of BC. Therefore, AD divides the triangle ABC into two smaller congruent triangles (ABD and ACD). The perpendicular bisector divides the hypotenuse into two segments, with the segments in the same ratio as the legs of the original right triangle. Therefore, using the Pythagorean theorem to find BC, we get BC = 24. Since D bisects BC, BD = 12. Applying the Pythagorean theorem to triangle ABD, we get AD = 13.

The incircle of a triangle is a circle that is tangent to all three sides of the triangle. The incenter of a triangle is the center of the incircle and it is the intersection point of the angle bisectors. It is known that the length of the tangent from a point to a circle is equal from all points on the circle. If the incircles of triangles ABD and ACD touch AD at a common point E, then the tangents from D to the incircles of both triangles are equal. Therefore, applying the tangent-tangent property, we can conclude that BD = CD.

In an isosceles triangle ABC with a right angle at B, a square is inscribed with three vertices on the sides of the triangle. The ratio of the side of the square to the base of the isosceles triangle is 2:1. Finding the area of the square and the triangle, we get the ratio of their areas as 2:5.

In triangle ABC, sides AB, AC, and BC are extended till Q, P and R such that AC = AP, BC = CR, and AB = BQ. This creates three new isosceles triangles: ACP, BCR, and ABQ. Since the area of triangle ABC is 10 sq cm, and each new triangle has double the base and height, the area of each new triangle is 40 sq cm. The area of Triangle PQR is simply the sum of the areas of these new triangles plus the area of the original triangle. Therefore, the area of triangle PQR is 40 + 40 + 40 + 10 = 130 sq cm.

When the water lily bends at the bottom of its stem, it creates a right triangle with the stem as the hypotenuse, the distance from the point where it originally came out of the water to the point where it disappears as one leg, and the depth of the lake as the other leg. Using the Pythagorean theorem, we can solve for the depth of the lake, which is 5m.

In the figure, ABCD is a square and ABE is an equilateral triangle. The measure of each angle in an equilateral triangle is 60 degrees. Since AB is a common side to both the square and the equilateral triangle, angle ABE is 60 degrees. Angle ABC is 90 degrees as it is an angle of the square. Therefore, angle CBE is 90 - 60 = 30 degrees. Similarly, angle DEC is also 30 degrees.

In triangle ABC, D and E are any points on AB and AC such that AD = AE. This means triangle ADE is an isosceles triangle. Since AD = AE, angles AED and ADE are equal. The sum of angles in a triangle is 180 degrees. Therefore, angle DAE = 180 - 2 * angle AED. Therefore, angle DAE is twice the measure of angle AED, which means that AD is the angle bisector of angle A.

If two wires are completely taut and cross each other, they form a right angle at the point of intersection. The height at which they cross is the perpendicular distance from the point of intersection to the ground. In this case, the two wires are represented as 'a' and 'b'. Because of similar triangles, the height at which the wires cross is equal to

2ab
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a + b