📐 geometry

1. **Problem statement:** Find the vertices of the triangle invariant under two transformations:

1. Problem 14: Given a figure with BC = 4, AB = 6, AD = 3 units, and \(\angle AEC = 90^\circ\), find the length of EC. 2. Since \(\angle AEC = 90^\circ\), triangle AEC is a right t

1. The problem involves performing dilations on points or shapes with given centers and scale factors $k$. 2. The dilation formula for a point $P(x,y)$ with center $C(x_c,y_c)$ and

1. **Stating the problem:** We have a triangle with a perpendicular dropped from the top vertex to the base, splitting the base into two segments of lengths 50 and 46 units. The si

1. مسئله: تعداد مکعب‌های موجود در تصویر را پیدا کنیم. 2. برای حل این مسئله، باید مکعب‌ها را به صورت جداگانه بشماریم و توجه کنیم که مکعب‌های بزرگ‌تر ممکن است از ترکیب چند مکعب کوچک‌

1. مسئله: تعداد مکعب‌های موجود در تصویر را بیابیم. 2. برای حل این مسئله، باید تعداد مکعب‌های کوچک، متوسط و بزرگ را جداگانه بشماریم و سپس جمع کنیم.

1. مسئله: تعداد مکعب‌ها را پیدا کنید. 2. برای محاسبه تعداد مکعب‌ها، باید بدانیم مکعب‌ها چگونه چیده شده‌اند یا ابعاد ساختار مکعبی چیست.

1. Statement of the problem: Find the area and centroid of the quadrilateral with vertices $D(5,4)$, $E(8,2)$, $F(7,-2)$, and $G(4,-4)$. 2. Formulas and rules used: Use the shoelac

1. The problem asks for the coordinates of point C on the given polygon. 2. From the graph description, the coordinates of the vertices are given explicitly.

1. **State the problem:** We have two shapes made from identical semicircles. The first shape has three semicircles with a total perimeter of 36 cm. We need to find the perimeter o

1. **State the problem:** We have a shape made of three identical semicircles with a total perimeter of 36 cm. We need to find the perimeter of a new shape made from the same semic

1. **State the problem:** We need to find the bearing of point Y from point W given the angles between rays from W. 2. **Understand bearings:** Bearings are measured clockwise from

1. We are asked to find the arc length of a partial circle with radius $7$. 2. The formula for the arc length $L$ of a circle segment is:

1. **State the problem:** We are given the circumference of a circle as 452.16 units and need to find the radius. 2. **Formula used:** The circumference $C$ of a circle is related

1. **State the problem:** We are given the circumference of a circle as 113.04 units and need to find the diameter. 2. **Formula used:** The circumference $C$ of a circle is relate

1. **State the problem:** We need to find the circumference of a circle with radius $3$ units. 2. **Formula:** The circumference $C$ of a circle is given by the formula:

1. **State the problem:** We are given the diameter of a circle as $16$ units and need to find its circumference. 2. **Formula:** The circumference $C$ of a circle is given by the

1. Find the value of $x$ that makes $m \parallel n$. **a.** Given angles: $100^\circ$ and $2x^\circ$ are corresponding angles.

1. **Problem Statement:** If two parallel lines are cut by a transversal, identify which pairs of angles are congruent and which are supplementary.

1. **Problem Statement:** If two parallel lines are cut by a transversal, identify which pairs of angles are congruent and which are supplementary.

1. The problem states the angles of a quadrilateral are in the ratio 3:4:5:6. We know the sum of angles in any quadrilateral is $360^\circ$. 2. Let the common ratio be $x$. Then th