📐 geometry

1. **Problem:** Find the area of isosceles triangle ΔPQR where PE ⊥ QR, PE = 1.6 cm, QR = 5.5 cm. Step 1: Since PE is perpendicular to QR, PE is the height of the triangle.

1. **State the problem:** Calculate the surface area (excluding the top rectangle) and volume of a half cylinder with radius $r=34$ cm and length $l=0.58$ m. Convert all units to c

1. **State the problem:** We have a triangle with points B, C, G where \(\triangle BCG\) is isosceles with \(|BC| = |CG|\) and \(\angle BCG = 36^\circ\). Line segment \(BD\) is par

1. The problem states that we have two similar triangles with areas 140.8 cm² and 178.2 cm², and we need to find the ratio of their corresponding medians. 2. For similar triangles,

1. **State the problem:** We are given a diagram with angles at point Q. We know \(\angle PQR = 90^\circ\), \(\angle PQR = 32^\circ\) (likely a typo, assume \(\angle PQS = 32^\circ

1. **State the problem:** We have two similar triangles with corresponding angle bisectors of lengths 3 and 5. The area of the smaller triangle is 45. We need to find the area of t

1. The problem is to show the ratio of the sides of a triangle or geometric figure. 2. Typically, side ratios are expressed as fractions or decimals comparing the lengths of sides.

1. **State the problem:** We have two chords AC and BD intersecting at point O inside a circle. We need to find the ratio of the areas of triangles AOB and COD. 2. **Recall propert

1. **Problem statement:** We have three similar triangles with side lengths in the ratio $3:4:5$. The sum of their areas is 300 cm². We need to find the area of the smallest triang

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