📐 geometry

1. Problem: Analyze the ellipses in Figures 1 and 2 and find the center, vertices, co-vertices, foci, lengths of axes, and equations for each. **Figure 1:**

1. **Stating the problem:** We have three power stations K, L, and M with the following information:

1. Show that [MR] and [AS] are the perpendicular bisectors of each other in square MARS. Since MARS is a square, all sides are equal and angles are $90^\circ$.

1. Stating the problem: Find the distance from the point $(2,3)$ to the straight line given by the equation $$4x + 3y = 10$$. 2. Formula for the distance $d$ from a point $(x_0, y_

1. Stating the problem: Find the cross-sectional area, lateral surface area, total surface area, and volume for each prism (a) through (च). 2. Prism (a): Rectangular prism with len

1. **Rectangular prism (a): length = 10 cm, width = 6 cm, height = 8 cm** - Cross-sectional area (base area): $A = \text{length} \times \text{width} = 10 \times 6 = 60 \text{ cm}^2

1. **Problem (a):** Rectangular prism with length $10$ cm, width $6$ cm, height $8$ cm. - Cross-sectional area (base area) $= \text{length} \times \text{width} = 10 \times 6 = 60$

1. **Phát biểu bài toán:** Cho tam giác đều ABC và các điểm M, N, P trên các cạnh BC, CA, AB sao cho - BM = k \cdot BC

1. Problem: Given square ABCD is congruent to square PQRS and side AB = 4 cm, find the length of side QR. Since congruent squares have equal corresponding sides, QR = AB = 4 cm.

1. **Stating the problem:** A semi-circle has a perimeter of 100 m. We want to find the length of the diameter in centimeters. 2. **Understanding the perimeter of a semi-circle:**

1. The problem states that a semi-circle has a perimeter of 100 m and asks to find the length of the diameter in cm. 2. The perimeter $P$ of a semi-circle consists of the diameter

1. **State the problem:** We have an open rectangular box with length $1$ m, width $70$ cm, and depth $50$ cm. 2. **Convert all dimensions to meters:**

1. مسئله داده شده این است که نقطه A با مختصات (7,6) راس یک متوازی الاضلاع است که دو ضلع آن روی خطوط $$3x - 2y = 11$$ و $$4x + 3y = 8$$ قرار دارند. باید مختصات وسط قطر این متوازی ال

1. **Stating the problem:** We are given two angles that are supplementary. This means their sum is $$180^\circ$$. 2. Let the smaller angle be $$x$$. Then the larger angle is three

1. Let's state the problem: A regular hexagon has several lines of symmetry, and we want to find how many lines of symmetry it has and the angle between any two adjacent lines of s

1. **Problem statement:** We have a circle with center $O$ and a tangent line $\overleftrightarrow{AC}$ at point $C$ on the circle. 2. Given that $\angle BAC = 23^\circ$, and since

1. The problem asks for the area of a quarter circle with radius $4$ units. 2. The area of a full circle is given by the formula $$A=\pi r^2$$ where $r$ is the radius.

1. The problem asks for the area of a semicircle with radius 6 units. 2. The formula for the area of a full circle is $$A = \pi r^2$$ where $r$ is the radius.

1. **State the problem:** Given that angle A is circumscribed about circle O, find the measure of \(\angle A\) given that the central angle \(\angle BOC = 92^\circ\). 2. **Recall t

1. The problem gives a triangle ABC with sides AB = 24 units, AC = 10 units, and BC = 5 units. 2. The circle P inside the triangle is tangent to all three sides, meaning it is the

1. The problem states that $\angle AOC$ is a central angle and intercepts arc $\widehat{AC}$. The measure of a central angle is equal to the measure of the intercepted arc.\n\n2. G