📐 geometry

1. **Problem Statement:** Given an isosceles trapezoid front face of a sign with sides WX = $2x - 2$, YZ = $2x + 6$, WZ = $4x + 5$, and XY = $5x - 3$, prove $x=8$, find $m\angle Z$

1. **Stating the problem:** We are given a rectangular prism with dimensions 4 cm (height), 12 cm (length), and 3 cm (width). We need to find the length of the diagonal inside the

1. **State the problem:** We are given a circle with points P, Q, R, S on the circumference and angles 126°, 93°, and 90° at certain points. We need to find the measures of angles

1. **Problem statement:** Find the angle $\angle HFE$ in the given geometric figure. 2. **Identify known elements:** To find $\angle HFE$, we need information about points $H$, $F$

1. The problem asks: If you know the diameter of a circle, how can you find the area? 2. Recall the formula for the area of a circle: $$A = \pi r^2$$ where $r$ is the radius.

1. **State the problem:** Find the circumference of a circle with diameter 20 ft, using $\pi = 3.14$, and round to the nearest hundredth. 2. **Formula:** The circumference $C$ of a

1. **Stating the problem:** We need to find the measure of angle $HFE$ in the polygon with vertices $H, E, G, F$. Given are sides $HE=38$ cm, $EG=35$ cm, $GF=60$ cm, angles $\angle

1. Problem (a)(i): Construct the perpendicular bisector of chord [AB] using only a compass and straight edge. - To construct the perpendicular bisector of [AB], place the compass a

1. **Problem statement:** We have two intersecting lines AB and CD at point E.

1. **Problem statement:** We have two intersecting lines AB and CD at point E.

1. **Problem statement:** (a)(i) Given the circle equation $x^2 + y^2 = 49$, find the centre and radius.

1. **State the problem:** We need to find the area of the yellow shape formed by two intersecting circles where the angle between the two radii is 100° and the distance between the

1. **State the problem:** We know that $\frac{3}{8}$ of the rectangle is shaded, and the shaded area is 72 m$^2$. The width of the rectangle is 12 m. We need to find the height of

1. **Problem statement:** We have a polygonal side view of a building with given side lengths and angles: $v=10^\circ$, $u=40^\circ$, and we need to argue that $w=40^\circ$, then f

1. **Problem statement:** Calculate the perimeter of the shaded region and the volume of wood to be cut off from the wooden door stopper. 2. **Given:**

1. **Stating the problem:** We have a sequence of right triangles where the shortest side $a=1$ is fixed. The triangles are:

1. **Stating the problem:** Calculate the area ($a$) and perimeter ($p$) of the right triangle in the middle-right position with sides 6 cm, 8 cm, and hypotenuse 10 cm.

1. **State the problem:** We need to find the volume of a shape made by combining a cone and a hemisphere. The hemisphere has radius $r=7$ cm, and the cone has radius $r=7$ cm and

1. **Énoncé du problème :** On a les points $A(-2;3)$, $B(4;1)$, et $C(3;-2)$.

1. **Problem statement:** We are given a triangle ABC with sides AC = $7\sqrt{3}$ cm, AB = $\sqrt{6}$ cm, BC = $a$ cm, and angle $\angle A = 45^\circ$. We want to show that $a = \s

1. **State the problem:** We need to find the coordinates of the image R''S''T''U'' after reflecting the figure RSTU across the x-axis. 2. **Recall the reflection rule across the x