📐 geometry

1. Problem 4.1: In \(\triangle DEF\), \(\angle E = 90^\circ\), \(DE = 15\) m, and \(DF = 17\) m. Find the length of \(EF\). 2. Use the Pythagorean theorem for right triangles: $$DF

1. **Problem Statement:** We have a cube whose side length increases by 10%. We need to find:

1. Let's start by stating the problem: We want to understand what area means in mathematics and how to calculate it. 2. Area is the measure of the amount of space inside a two-dime

1. **Problem Statement:** Calculate the area of the irregular polygon with given side lengths 6 ft, 3 ft, 8 ft, 9 ft, 6 ft, 13 ft, 6 ft, and 8 ft. 2. **Approach:** To find the area

1. **State the problem:** Find the area of the irregular polygon shaped like an arrow using the given side lengths. 2. **Approach:** The polygon can be divided into three rectangle

1. **Problem Statement:** Find the area of the irregular polygon with given side lengths: 15 mm (left vertical), 9 mm (bottom horizontal), 6 mm (right vertical), and smaller segmen

1. **State the problem:** Find the area of the irregular polygon composed of rectangles and triangles with given side lengths. 2. **Analyze the figure:** The polygon can be divided

1. **State the problem:** We need to find the area of a compound figure composed of rectangles and a triangle with given side lengths. 2. **Break down the figure:** The figure can

1. The problem asks to identify which pairs of triangles are congruent by ASA (Angle-Side-Angle). 2. ASA congruence means two triangles are congruent if two angles and the included

1. The problem asks what additional information is needed to prove that triangle $\triangle ABC$ is congruent to triangle $\triangle DEF$ by the ASA (Angle-Side-Angle) criterion. 2

1. The problem asks which theorem is NOT valid to prove that two triangles are congruent. 2. The common triangle congruence theorems are:

1. **State the problem:** We need to determine which triangle congruence theorem proves that triangle $\triangle ABC$ is congruent to triangle $\triangle DEF$ given that side $BC =

1. **Problem Statement:** We have two right triangles, \(\triangle ABC\) and \(\triangle DEF\). \(\triangle ABC\) has legs \(y\) (vertical) and \(x\) (horizontal), and hypotenuse 1

1. **State the problem:** We need to determine which given information can be used to prove that triangle ABC is congruent to triangle DEF. 2. **Given information:**

1. **Problem Statement:** Determine which theorem can be used to show that triangle ABC is congruent to triangle DEF given that side AB equals side DE, side AC equals side DF, and

1. **Problem Statement:** We have an isosceles triangle RST with RS = ST. Point M(2,0) is the midpoint of RT, and \(\angle RMO = 45^\circ\). Point S has coordinates \((k,5)\). We n

1. **Problem Statement:** We have a parallelogram ABCD with sides AB = 32 mm, AD = 40 mm, and angle BAC = 77°.

1. The problem asks which pair of angles has a sum of 180° given two parallel lines PQ and RS cut by a transversal. 2. When two parallel lines are cut by a transversal, pairs of an

1. **Problem statement:** We are given a triangle with one angle of 30° and two sides: the side opposite the 30° angle is 2.2 cm, and the side adjacent to the 30° angle is 3.5 cm.

1. **Problem statement:** We have two similar parallelograms A and B. Parallelogram A has sides 3 cm and $y$ cm. Parallelogram B has sides 12 cm and 20 cm. We need to find the leng

1. **State the problem:** We need to find the length of side $x$ in a triangle where one side is 6.2 cm, and the angles adjacent to this side are 65° and 51°. 2. **Identify the kno