📐 geometry

1. **Problem statement:** We have triangle MNR with angle $M = 120^\circ$, side $MN = 9$, side $NR = 16$, and we need to find angle $x = \angle R$. 2. **Formula used:** Use the Law

1. **State the problem:** Calculate the area of the triangles outlined by dotted lines. 2. **Formula for the area of a triangle:** The area $A$ of a triangle is given by

1. **Problem:** Find the circumference of a circle with diameter 16 cm in terms of $\pi$. 2. **Formula:** The circumference $C$ of a circle is given by

1. **State the problem:** Calculate the area of a triangle with base 20 cm and height 12 cm. 2. **Formula:** The area $A$ of a triangle is given by the formula:

1. **State the problem:** Given triangle ABC with angles $\angle A = 32^\circ$, $\angle C = 76^\circ$, and side $a = 9$ opposite angle A, find the lengths of sides $b$ and $c$. 2.

1. **State the problem:** We have a rectangle QUAD with points Q(0,0), U(0,3), A(4,3), and D(4,0). The rectangle is dilated from center (0,0) with scale factor 5. We need to find t

1. **Problem:** Find the area and circumference of a circle with radius 2 ft. 2. **Formulas:**

1. **Problem statement:** We have triangle A-C-B with vertex C at the top, base AB horizontal, side AC labeled $b$, side CB labeled $a=8$, and base AB labeled $c$. Given angles $\a

1. **Problem Statement:** We are given two triangles, △FGH and its image △NOP after a dilation. We need to determine if the scale factor of the dilation is greater than or less tha

1. **State the problem:** We have two parallelograms, a smaller one with vertices Q(1,1), R(3,1), S(4,3), T(2,3) and a larger one with vertices Q'(4,3), R'(9,3), S'(12,9), T'(6,9).

1. **State the problem:** We are given two angles formed by intersecting lines: one angle is 62° and the opposite angle is labeled as $(x - 2)^\circ$. We need to find the value of

1. **State the problem:** We are given two adjacent angles on a straight line: one angle is $(x + 2)^\circ$ and the other is $57^\circ$. We need to find the value of $x$. 2. **Form

1. **State the problem:** Find the area of a regular hexagon with a given apothem (distance from center to a side) or radius (distance from center to a vertex) of $12\sqrt{3}$ cm.

1. **State the problem:** Find the area of a regular polygon given the radius (or apothem) of 15 cm. 2. **Formula for the area of a regular polygon:**

1. **State the problem:** We are given a triangle with vertices A, B, and C. Given:

1. **State the problem:** We have a triangle with sides 70, 52, and 48 units. We want to find the angle $x$ opposite the side of length 70. 2. **Formula used:** The Law of Cosines

1. **Calculate the distance around the running track.** The running track is a rectangle with two semicircles at the ends.

1. **State the problem:** We have two mathematically similar cuboids. The smaller cuboid has a shortest edge of 5 cm and a surface area of 400 cm^2. The larger cuboid has a shortes

1. **State the problem:** We have two similar rectangles ABDE and BCFA. We know the dimensions of BCFA are 15 m (height) and 7.5 m (width). We need to find the length of wall FE in

1. **State the problem:** We need to find the weight of a wooden headboard shaped like a house with given dimensions.

1. Problem 57: Find the measure of angle at U in kite XUVW where \(\angle W = 87^\circ\). 2. In a kite, two pairs of adjacent sides are equal, and one pair of opposite angles are e