📐 geometry

1. **State the problem:** We need to find the ratio of the area of a sector with central angle $\alpha$ radians to the area of the circle inscribed in that sector. 2. **Recall form

1. The problem is to draw a triangle, which is a geometric figure with three sides and three angles. 2. Since this is a drawing request and not a mathematical problem, we cannot pr

1. The problem asks for the radius of a circle given its circumference of 135 cm. 2. The formula for the circumference $C$ of a circle is:

1. State the problem: Given an isometric sketch of a solid, produce its oblique sketch using an oblique projection method. 2. Formula and projection rule:

1. **Problem statement:** Prove that the square ABCD and the rectangle CMEF have equal areas given that CF = CP and \(\angle CBP = \angle BMP\).

1. **Problem statement:** A rectangle is divided into four smaller rectangles. Three of them have areas 6 cm², 18 cm², and 36 cm². We need to find the area of the shaded rectangle.

1. **Problem statement:** We have two chords WF and HG intersecting inside a circle at point E. Given angles are $\angle W = 55^\circ$ and $\angle G = 71^\circ$. We need to find th

1. **Problem statement:** Find the measure of the unknown arc or angle indicated in the circle with given angles 52° and 121°. 2. **Relevant formula:** In a circle, the sum of angl

1. **Problem statement:** Calculate the surface area of a cylinder. 2. **Formula:** The total surface area $A$ of a cylinder is given by

1. **State the problem:** We need to determine the placement of 6 sprinklers to cover a 9 by 16 tile plot, where each sprinkler waters 24 adjacent tiles. 2. **Understand sprinkler

1. **Stating the problem:** We have two triangles \(\triangle PQR\) and \(\triangle VUT\) that are similar (\(\triangle PQR \sim \triangle VUT\)). We know the angles and some side

1. **State the problem:** We need to find the perimeter and area of a rectangle with vertices at (5, -1), (5, -5), (-3, -5), and (-3, -1). 2. **Recall formulas:**

1. **State the problem:** Find the area and perimeter of the rectangle with vertices $(-3, 3)$, $(-3, -7)$, $(4, -7)$, and $(4, 3)$. 2. **Identify the lengths of the sides:** Since

1. **State the problem:** We are given a triangle with points B, K, C, L, and D. Segments BK = 10, KC = 5, BL = 13, and LD = y. We need to find the length of segment LD, denoted as

1. **Problem statement:** We are given a geometric figure with points B, K, C, L, and D. The segments BK = 10, KC = 5, BL = 13, and LD = y. We need to find the length y. 2. **Under

1. The problem asks if the lighter colored quadrilateral is a dilation of the smaller quadrilateral. 2. A dilation means the shapes are similar: all corresponding angles are congru

1. **Problem Statement:** Given that ABCD is a parallelogram, determine which of the following statements are NOT true: - Diagonals are congruent in length

1. **Problem 1A: Reflecting point P over the x-axis** The original point P is at coordinates $P(3,5)$. When reflecting a point over the x-axis, the x-coordinate remains the same, b

1. **State the problem:** Calculate the surface area of the attic given by the expression $$45 (40 + 25 + 25) + \frac{1}{2} (40 \times 15)$$. 2. **Understand the formula:** The exp

1. **State the problem:** We need to determine which shortcut can be used to prove that the two right triangles shown are congruent. 2. **Identify given information:** Both triangl

1. **State the problem:** Find the distance between each pair of points given. 2. **Formula used:** The distance $d$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ in the plane i