📐 geometry

1. **State the problem:** We need to find the length of side $c$ in a triangle where sides $a=8$, $b=14$, and angle $C=73^\circ$ are given. 2. **Formula used:** The Law of Cosines

1. **State the problem:** We need to express the area $A$ of a circle in terms of its diameter $d$. 2. **Recall the formula for the area of a circle:** The area is usually given by

1. **State the problem:** We need to find the area of a composite figure composed of rectangles and a triangle. 2. **Analyze the figure:** The figure consists of:

1. **State the problem:** We are given a triangle with vertices $A(1,2)$, $B(4,2)$, and $C(2,5)$.

1. **Stating the problem:** We have a right triangle with legs labeled $3x$ and $4x$, and the hypotenuse labeled $3$. We need to find the value of $x$. 2. **Formula used:** In a ri

1. **State the problem:** We have a straight line $E D$ with point $A$ between $E$ and $D$. Rays $A B$ and $A C$ create three angles: $E A B = 100^\circ$, $B A C = x^\circ$, and $C

1. **Problem Statement:** We have an isosceles triangle $OAB$ with $OA = OB$ and angle $AOB = \theta$ radians. Points $C$ and $D$ lie on $OA$ and $OB$ respectively. $CD$ is an arc

1. **Problem Statement:** We have two right triangles.

1. The problem is to find the volume of a given solid. 2. The formula for the volume depends on the shape of the solid. Common formulas include:

1. 問題の確認：△ABCにおいて、AB=4、AC=5、∠A=60°です。まず、(1) △ABCの面積を求めます。 2. 面積の公式：三角形の面積は2辺の長さとその間の角度を使って求められます。

1. **State the problem:** Find the distance between points $X(-2,4)$ and $Y(3,-2)$ on the coordinate plane. 2. **Formula used:** The distance $d$ between two points $(x_1,y_1)$ and

1. **State the problem:** We need to find the lengths of the legs $XZ$ and $YZ$ of right triangle $XYZ$ with coordinates $X(-2,4)$, $Z(3,4)$, and $Y(3,-2)$. 2. **Recall the formula

1. The question asks if we can use the Pythagorean theorem. 2. The Pythagorean theorem states that in a right triangle, the square of the hypotenuse $c$ is equal to the sum of the

1. **State the problem:** We need to find the length of side $q$ in triangle $PQR$ where $PQ=10$ cm, $QR=14$ cm, and the angle at $Q$ is $98^\circ$. 2. **Identify the sides and ang

1. **Problem Statement:** Determine the number of lines of reflective symmetry for a shape formed by overlapping two congruent diamonds (a 4-point star).

1. **Problem Statement:** Reflect the letter $Y$ first across the vertical line (y-axis) and then across the horizontal line (x-axis).

1. **Problem statement:** Given a map scale of 1 : 5000, find the actual distances corresponding to map distances of 4 cm and 7 cm.

1. **Problem:** Find the angle between the plane defined by points $B$, $G$, $D$ and the plane $ABCD$ of a cube with side length $10$ cm. 2. **Understanding the problem:** The cube

1. **State the problem:** We need to find the value of $x$ in a circle where a radius is drawn perpendicular to a chord of length $23$, and the perpendicular segment from the cente

1. **State the problem:** We need to find the value of $x$, the length of segment $PO$, given the geometric conditions and lengths in the circle.

1. **Problem Statement:** Draw triangle $PQR$ where side $PQ = 4.5$ cm, side $PR = 11.7$ cm, and angle $\angle PQR = 90^\circ$. 2. **Understanding the problem:** We have two sides