📐 geometry

1. **State the problem:** We need to find the area of the shaded sector XPY of a circle with radius 5 cm and central angle 60°. 2. **Formula for the area of a sector:** The area $A

1. **Stating the problem:** We have two intersecting lines forming a right angle (90°) and an angle of 68° marked between the lower left and upper right arms. 2. **Identify the ang

1. **Problem 1: Find the area of the irregular polygon (in inches).** The figure is composed of a rectangle and a right triangle on top.

1. **State the problem:** We need to find the area of triangle $\triangle UVW$ where side $UV = 41$ cm, side $VW = 48$ cm, and the angle at vertex $V$ is $64^\circ$. 2. **Formula u

1. **State the problem:** We have a large rectangle of width 7 meters and height 6 meters. Inside it, there is a smaller rectangle at the top-left corner measuring 1 meter wide and

1. **Stating the problem:** We have a composite shape made of two rectangles joined side-by-side. Each rectangle is 7m by 8m, but there is a 1m by 1m cut or overlap in the top-left

1. **Problem statement:** Given a circle with center $C$ and radius $41$ mm, a tangent line $QP$ at point $Q$ on the circle with length $QP=9$ mm, find the length of chord $CS$. 2.

1. **Nêu bài toán:** Chứng minh ba điểm I, M, K thẳng hàng, với I trên AC, K trên DB, và AI = DK.

1. **Problem statement:** Given triangle ABC with angles $\hat{A} = 80^\circ$ and $\hat{B} = 50^\circ$, prove that triangle ABC is isosceles. 2. **Step 1: Find the third angle $\ha

1. **Stating the problem:** Find the image of point $A(-5,3)$ reflected about the line $x=-2$. 2. **Reflection rule about vertical line $x = a$:** The $x$-coordinate of the image i

1. **State the problem:** Calculate the area of a sector of a circle with radius $20$ cm and central angle $300^\circ$. 2. **Formula:** The area $A$ of a sector with radius $r$ and

1. **State the problem:** We have a right triangle with hypotenuse 50 ft and legs 30 ft and 40 ft. A height $h$ is drawn from the right angle to the hypotenuse, dividing the base i

1. **Problem Statement:** Calculate the perimeter of the polygon with sides 4 cm, 8 cm, and 12 cm. 2. **Formula:** The perimeter $P$ of a polygon is the sum of the lengths of all i

1. The problem states that quadrilateral LMNP is similar to quadrilateral L'M'N'P'. We need to identify which equations correctly represent the ratios of corresponding sides to sol

1. **Problem statement:** Given that segment $A'C'$ is parallel to segment $AC$, find the length of segment $AC'$. Known lengths are $AA' = 6$, $A'B = 8$, $BC' = 10$, and $CC' = 5$

1. **State the problem:** We need to find the area of a circle with radius $r = 1$ yard. 2. **Formula for the area of a circle:** The area $A$ is given by the formula

1. The problem involves finding unknown angles or lengths in a geometric figure with a circle and star shape, where angles 60° and 70° are given at point S. 2. To solve problems in

1. The problem is to find the location of the point with coordinates $(2,3)$ on the Cartesian coordinate plane. 2. The Cartesian plane consists of two perpendicular number lines: t

1. **Problem Statement:** Determine if triangle BED is equilateral given that angles at B and E are 60° and markings indicate equal sides. 2. **Recall:** An equilateral triangle ha

1. **Problem statement:** Determine if triangle BED is equilateral given that angles at B and E adjacent to diagonal BE are 60° each and sides BD and DE are equal. 2. **Given:** \(

1. Problem 19: Show that points (-1, -1), (2, 8), and (5, 17) are collinear. 2. To prove collinearity, we check if the slope between each pair of points is the same.