📐 geometry

1. **State the problem:** Given a triangle with sides $a=20$, $b=18$, and angle $C=75^\circ$, find side $c$ using the Law of Cosines. 2. **Formula:** The Law of Cosines states:

1. **Problem statement:** We have two intersecting circles VWZ and WXYZ intersecting at points W and Z. Given:

1. **Problem statement:** We have two intersecting circles VWZ and WXYZ intersecting at points W and Z. Given:

1. Given a triangle with a vertical left side of length 9, a horizontal base of length 10, and the left angle of 50°. 2. Use the Law of Cosines or trigonometric ratios to find the

1. The problem asks which angle measure among $w$, $x$, $y$, and $z$ should equal 92° to prove that lines $r$ and $s$ are parallel. 2. When two lines are cut by a transversal, corr

1. **State the problem:** We have two lines $f$ and $g$ crossed by a transversal $n$. We want to find which angle measure among $a$, $b$, $c$, and $d$ should equal $105^\circ$ to p

1. **State the problem:** Find the area between the two circles defined by the equations $$y^2 + x^2 = 4$$ and $$x^2 - 4x + y^2 = -3$$ for $$1 \leq x \leq 2$$. 2. **Rewrite the sec

1. The problem involves translating geometric shapes on a vertical grid. 2. Translation means moving a shape without rotating or resizing it. Moving "1 place right, 3 places down"

1. **Problem statement:** Calculate the missing size of the circular ring given inner radius $r_i$, outer radius $r_a$, and area $A$ of the ring. 2. **Formula:** The area of a circ

1. **State the problem:** We have a right triangle with two legs given: one leg is 3 units and the other leg is 7 units. We need to find the length of the hypotenuse (the side oppo

1. The problem asks to identify the figure shown and name it using the given points X and Z. 2. The figure is described as a line segment with endpoints X and Z.

1. **Problem statement:** Given triangle ABC with angles $\angle C = 58^\circ$, $\angle A = 93^\circ$, and side $AC = 16$, find the length of side $BC$. 2. **Step 1: Find the missi

1. Xác định điểm M, N lần lượt là trung điểm của AE và BC. 2. Ta cần chứng minh DE vuông góc với MN.

1. The problem is to find the center of a circle given its equation or a geometric figure. 2. The center of a circle is the point equidistant from all points on the circle.

1. **Problem stated:** You asked how to do a VIN diagram, and whether I can draw it too. 2. A **Venn diagram** is a picture used to show how groups overlap.

1. **Stating the problem:** We are given three geometric figures with various side lengths labeled. The first figure is a cone with a slanted side of 30 cm and a base diameter of 2

1. **Problem 1: Triangle E-D-F** Given: side $ED=20$, angle $E=44^\circ$, angle $F=80^\circ$. Find missing side $DF$ and angle $D$.

1. **State the problem:** We need to find the size of angle $c$ in a figure with two triangles sharing a middle segment, given angles $20^\circ$ and $62^\circ$, and two equal-lengt

1. **State the problem:** Find the volume of a cylinder with radius $r=7$ cm and height $h=20$ cm. 2. **Formula:** The volume $V$ of a cylinder is given by

1. The problem asks to construct a line perpendicular to a given line segment passing through a specific point. 2. To construct a perpendicular line, we use the rule that perpendic

1. **Stating the problem:** We have a circle with points A, B, D, and a chord AD. Given that segment AB = 3 cm, BD = X, and X = P, we want to understand the relationships between t