📐 geometry

1. **State the problem:** We have a square PQRS with diagonal PR. Points P and R have coordinates $P(4,7)$ and $R(8,-5)$. We need to find the equation of the line passing through p

1. **State the problem:** We need to find the area of a trapezoid with bases of lengths 12 cm and 6 cm, and a height of 9 cm. 2. **Recall the formula for the area of a trapezoid:**

1. **State the problem:** We need to find the volume of an oblique cylinder with a base radius of 4 m and a height of 10 m. 2. **Recall the formula for the volume of a cylinder:**

1. **State the problem:** Find the distance between points $E(-4, 3)$ and $F(-4, -3)$.\n\n2. **Recall the distance formula:** The distance $d$ between two points $(x_1, y_1)$ and $

1. **State the problem:** We have a circle centered at point $O$ with tangent segments $DE$ and $DF$ from point $D$ to the circle. Given $OE = 8.1$ and $OD = 13.5$, we need to find

1. **State the problem:** We have points A, X, and B on a grid. We want to find the column vectors from X to A, then from X to A' and X to B' after enlargement by a scale factor of

1. The problem asks for the coordinates of vertices A and B after reflecting the shape about the line $y = x$. 2. Reflection about the line $y = x$ swaps the $x$ and $y$ coordinate

1. Problem: Prove triangle HGE is congruent to triangle FGE given that segment GE is an angle bisector of both angle HEF and angle FGH. Step 1: Since GE bisects angle HEF, we have

1. **State the problem:** We have a triangle ABC with angles \(\angle ACB = 110^\circ\), \(\angle CAB = \theta^\circ\), and \(\angle ABC = f(\theta)\). We need to express \(f(\thet

1. **State the problem:** We need to find the angle $\angle MLN$ in triangle $MLN$ where $\angle M = 62^\circ$, side $MN = 15$ cm, and side $LN = 20$ cm. 2. **Identify what we know

1. **State the problem:** We need to find the angle $\angle MLN$ in triangle $MLN$ where $ML=20$ cm, $MN=15$ cm, and $\angle M=62^\circ$. 2. **Identify known values:**

1. The problem asks to identify the theorem, term, or corollary represented by a triangle with two sides marked as equal (bold sides) and no angles marked. 2. The bold sides indica

1. The problem asks to identify the theorem, term, or corollary represented by the congruence of triangles \(\triangle ABC \cong \triangle DEF\) and the markings shown. 2. Given \(

1. **State the problem:** We need to find the surface area of a triangular prism with triangular base sides 13 cm, 16 cm, and 12 cm, and prism length (depth) 5 cm. 2. **Calculate t

1. **State the problem:** Find the lengths of the line segments \(\overline{CM}\) and \(\overline{AB}\) using the distance formula. 2. **Calculate \(CM\):**

1. **State the problem:** Prove that right triangles $\triangle ABD$ and $\triangle CBD$ are congruent. 2. **Find the length of segment $AD$ using the distance formula:**

1. **State the problem:** We are given right triangles \(\triangle ABD\) and \(\triangle CBD\) with points \(A(0,0)\), \(B(4,6)\), \(C(8,0)\), and \(D(4,0)\). We need to prove that

1. Let's first clarify the problem: we want to find the angle $\angle ABO$ in a given geometric figure where points A, B, and O are defined. 2. Typically, to find an angle like $\a

Problem: Find the mass of a cylindrical cake with radius 5.5 cm and height 11 cm, given that each cubic centimetre has mass 0.4 g, and give the final answer to 1 d.p. 1. The volume

1. **Stating the problem:** We have a shape composed of an ellipse and two straight parallel lines connecting semicircles at each end. The total length of the shape is 250 meters,

1. The problem asks how many complete cylinders can be made from a given material or volume. 2. To solve this, we need to know the total volume or material available and the volume