📐 geometry

1. 문제: 두 점 A(-1, a)와 B(a, 5) 사이의 선분 AB의 길이가 최소가 되도록 하는 a의 값을 구하시오. 2. 선분 AB의 길이 공식은 두 점 사이의 거리 공식으로 주어집니다:

1. **State the problem:** We are given a rectangle ABCD with points A(3,4) and B(5,4). One of the diameters of the circle circumscribing this rectangle is given by the line $4y = x

1. **Problem Statement:** Find the total surface area of a square-based pyramid with vertical height and slant height in the ratio 4:5 and volume 384 cm³.

1. **Stating the problem:** We are given a geometric figure with angles $x$, $y$, and $z$ and the equation $x + y = 63^\circ$. We need to find the value of angle $z$. 2. **Understa

1. مسئله: در مثلثی با اضلاع ۷، ۵ و ۳ واحد، دایرهٔ محاطی خارجی بر ضلع متوسط (که طول آن ۵ است) و امتداد دو ضلع دیگر مماس است. باید نسبت تقسیم نقطه تماس بر ضلع متوسط را بیابیم. 2. ابت

1. مسئله: در یک مثلث، ارتفاع‌ها به ترتیب 4، 3 و 2 واحد هستند. باید محیط دایره محاطی داخلی مثلث را پیدا کنیم. 2. فرمول‌ها و نکات مهم:

1. The problem is to simplify the expression $n(n-3)$ and understand the formula for the sum of interior angles of a polygon, $S = (n-2)180^\circ$. 2. First, simplify the expressio

1. **State the problem:** We need to prove that the line segment $\overline{HI}$ is congruent to the line segment $\overline{XW}$ by using the given triangles $WVX$ and $HJI$ and t

1. The problem asks: How many triangles are there in an octahedron? 2. An octahedron is a type of polyhedron with 8 triangular faces and 6 vertices.

1. The problem asks how to determine if pool ABCD is similar to pool EFGH using geometric transformations. 2. Similarity between two shapes means one can be obtained from the other

1. The problem states that triangle $\triangle ABC$ is dilated from point $A$ by a scale factor of $\frac{1}{2}$. We need to find the relationship between segment $D'E'$ (the image

1. The problem asks to draw the net of each figure and calculate its surface area. 2. For figure (a), a triangular pyramid with edges 8 cm, 6 cm, and 6 cm:

1. **State the problem:** We have a segment $IJ$ with length 2, and it is dilated by a scale factor of $\frac{3}{4}$ to form segment $I'J'$. We need to find the length of $I'J'$.\n

1. The problem states that the measure of angle $\angle HIJ$ is given as 94°. 2. It also states that $m \angle HIJ = 90^\circ$, which is a contradiction because an angle cannot be

1. The problem involves finding the missing side lengths in right triangles with given angles and side lengths, specifically using 30° and 60° angles. 2. For a 30°-60°-90° triangle

1. **Problem statement:** Given a right triangle ABC with a right angle at B, M is the midpoint of AC. From M, MH is drawn perpendicular to AB at H, and MK is drawn perpendicular t

1. The problem involves finding the value of $x$ given angles in a circle, where angles such as $x^\circ$, $2x^\circ$, $3x^\circ$, $-x^\circ$, and $130^\circ$ appear in various con

1. **Problem Statement:** Explain the properties of a quadrilateral, specifically focusing on the sum of interior angles and the diagonals forming triangles.

1. **Stating the problem:** We need to find the length of segment $JN$ by using the similarity of two right triangles $\triangle JKL$ and $\triangle JNM$. 2. **Identifying similar

1. **Problem Statement:** We need to find the length of segment $JN$ in the given right triangle configuration. 2. **Given:**

1. **State the problem:** We need to classify triangle $\triangle WXY$ with vertices $W(-7,1)$, $X(0,-4)$, and $Y(2,3)$ as equilateral, isosceles, scalene, or none of the above. 2.