📐 geometry

1. **Problem 1: Find angle PRQ in the triangle with angles expressed in terms of $x$.** Given angles:

1. **Problem:** Find the area of the L-shaped composite rectangle with outer dimensions 30 ft by 20 ft, a lower-left cutout with left vertical leg width 5 ft and right vertical dro

1. **State the problem:** Find the area of the composite shape which is an L-shaped rectangle with given side lengths. 2. **Formula and rules:** The area of a composite rectangle c

1. Staðfesta vandamálið: Við þurfum að finna rúmmál pýramídans sem hefur ferning sem botn með hliðarlengd $61$ cm og hliðar sem eru jafnliða þríhyrningar. 2. Formúla fyrir rúmmál p

1. Staðfesta vandamálið: Við höfum pýramída með ferningslaga botnflöt sem hefur hliðarlengd $124$ cm. 2. Gefið: Botnflötur er ferningslaga með hlið $124$ cm.

1. **State the problem:** We need to find the lateral surface area of a rectangular prism with dimensions 7.5 in. (length), 5 in. (width), and 6.5 in. (height). 2. **Formula for la

1. **Stating the problem:** We have a right triangle with legs labeled $a$ and $b$, and hypotenuse $c$. Given values are $b=4$, $a=6$, and we want to find $c$ using the Pythagorean

1. **Stating the problem:** We are given a right triangle with sides labeled $a$, $b$, and hypotenuse $c$. We want to verify the Pythagorean theorem formula:

1. **State the problem:** We have a rectangular fish tank with a bottom area of 450 inches² and it is filled with water to a height of 10 inches. We need to find the volume of wate

1. **Problem statement:** Find the incenter (intersection of angle bisectors) and the inradius (radius of the incircle) of triangle ABC with vertices A(0|2), B(7|3), and C(10|6). D

1. **State the problem:** We have triangle QRS with vertices Q(-1,9), R(4,4), and S(-3,1).

1. **Problem:** Find the measure of angle $\angle D$ in the parallelogram $DEFG$ where side $DG = 3.6$ ft, side $GF = 8$ ft, and $\angle F = 106^\circ$. 2. **Formula and rules:** I

1. **Find m\angle? in the trapezoid with one angle marked 112°** Given that trapezoid has one angle measuring 112° and the adjacent angle on the same base is m\angle?. Since these

1. **State the problem:** We are given a trapezoid with bases of lengths 2 cm and 4 cm, an area of 9 square centimeters, and a height $h$ to find. 2. **Formula for the area of a tr

1. **State the problem:** We need to find the height $h$ of a trapezoid with area 48 square miles, top base 6 miles, bottom base 10 miles, and one slanted side 7 miles. 2. **Formul

1. **State the problem:** We need to identify which three transformations among the given options prove that segment $AB$ is congruent to segment $A'B'$. Congruent segments have th

1. **State the problem:** We need to find the surface area of a spherical globe with diameter 12 inches. 2. **Recall the formula:** The surface area $S$ of a sphere is given by $$S

1. **State the problem:** We need to find the area of the shaded sector of a circle with radius 16 inches and central angle 150°. 2. **Formula for the area of a sector:** The area

1. The problem asks to find the expressions representing the two quantities David should use to approximate the amount of metal in a nut, which is a rectangular prism with a cylind

1. The problem asks to find the point equidistant from three buildings located at points F, G, and H. 2. In geometry, the point equidistant from all vertices of a triangle is the c

1. **State the problem:** We need to find the measure of angle $\angle C$ in triangle $ABC$ where $\angle A = 18^\circ$ and $\angle B = 110^\circ$. 2. **Recall the triangle angle s