📐 geometry

1. Problem: Two right triangles $\triangle PQR$ and $\triangle ABC$ are given. 2. In $\triangle PQR$ the right angle is at $Q$, so $\angle Q=90^\circ$.

1. **Problem statement:** In triangle ABC, AB = AC, and the angle bisector of \(\angle ABC\) intersects AC at D. Given that \(BC = BD = AD\), find the measure of \(\angle CBD\). 2.

1. a. Find $w$ given arcs 110° and 70° in a circle with inscribed angle $w$. - The inscribed angle theorem states: an inscribed angle equals half the measure of its intercepted arc

1. **Stating the problem:** Find the area of the shaded rectangular face of the 3D prism with dimensions 9 m, 5 m, and 7 m.

1. **Problem Statement:** Find the length of the segment $x$ in a circle where a chord of length 10.2 is given, a perpendicular segment from the chord to the circle center is 8, an

1. The problem is to find the total number of triangles in a large square subdivided into a 5x5 grid with diagonal lines from each corner and additional diagonals in the central 3x

1. **Problem Statement:** Given a triangle $\triangle ABC$ with $AB = BC$ and $\angle ABC = 60^\circ$, find the area of $\triangle ABC$. 2. **Known Information:**

1. Problem statement. A cuboid has length $12$, width $9$, and height $4$.

1. **Problem Statement:** Find the standard form of the equation of a circle given different conditions. 2. **Formula:** The standard form of the equation of a circle with center a

1. Problem statement: In the parallelogram $ABCD$, $AB=10$ cm, $AD=7$ cm and $\angle DAB=60^\circ$.\n 2. Diagram suggestion: Place $A$ at the origin, let $AB$ lie on the positive $

1. **Problem Statement:** We have a quadrilateral ABCD with opposite sides parallel (AB \parallel DC and AD \parallel BC). Angles at vertices A, B, D are $x^\circ$, $y^\circ$, and

1. **Problem statement:** Given triangle ABC with point D on side BC, angles \(\angle DAC = \angle BAD\), and lengths \(AC = 6.5\), \(CD = 3.4\), and \(AB = 4.9\), find the length

1. **Stating the problem:** We have a square with two diagonal lines intersecting inside it, creating two triangles. We know two angles: 40° at the top-left corner and 80° near the

1. **Stating the problem:** We have a square with four right angles (each 90°). Inside the square, there are angles 40°, 80°, and an unknown angle $x$°. 2. **Understanding the prob

1. **Problem Statement:** We have a square ABCD with vertices A(3,1), B(3,6), C(8,1), and D(8,6).

1. **Problem Statement:** Calculate the size of the marked angles in problem 2a, which is a square divided by two diagonals forming four angles labeled $\alpha$, $b$, $e$, and $d$.

1. The problem asks for the scalar factor given that the image length $A'$ is 15 units. 2. The scalar factor (scale factor) is the ratio of the image length to the original length.

1. **State the problem:** We have two similar triangles on the coordinate plane, triangle A with vertices at approximately $(-8,6)$, $(-2,6)$, $(-5,7.5)$ and triangle A' with verti

1. The problem is to understand how to find the scale factor between two similar figures. 2. The scale factor is the ratio of any two corresponding lengths in the two figures.

1. **Stating the problem:** We have two triangles, A and A', on the coordinate plane. Triangle A is smaller with vertices approximately at $(-5,6)$, $(-2,6)$, and $(-5,4)$. Triangl

1. **State the problem:** We have two right triangles inside a larger triangle. The left triangle has one leg of length 4 and a hypotenuse of length 10. The right triangle has one