📐 geometry

1. **State the problem:** We have a right triangle with one leg $x$, another leg 25 cm, and an angle of 48° opposite the 25 cm side. We need to find the length $x$ to the nearest c

1. **Problem Statement:** We have two bearings from the north direction: 135° and 060°, with distances 8 km and 15 km respectively. We want to find the relative position of the two

1. The problem asks to find the measure of segment AC. 2. To solve this, we need more information such as the length of AC or other related segments, or a figure with points A, B,

1. **Stating the problem:** We have a right-angled quadrilateral with given side lengths and an angle of 130° at point C. We need to find all missing measures, including side lengt

1. **Problem statement:** Given rhombus ABCD with \(\angle ADC = 64^\circ\) and equilateral triangle BCE, find \(\angle CAE\). 2. **Properties and formulas:**

1. The problem is to understand and apply CPCTC (Corresponding Parts of Congruent Triangles are Congruent). 2. CPCTC is used after proving two triangles are congruent by methods li

1. **State the problem:** Given that $\overline{CA} \cong \overline{CE}$ and $\overline{BA} \cong \overline{DE}$, prove that $\overline{BX} \cong \overline{DX}$. 2. **Identify know

1. **State the problem:** We need to find the total length of 12 vertical support beams on a bridge. There are 6 beams on each side, evenly spaced along the length of the bridge. 2

1. **State the problem:** Dilate figure PQRS by a scale factor of $\frac{1}{2}$ with the center of dilation at point P.

1. The problem asks to identify which pair of angles are alternate interior angles when two parallel lines are cut by a transversal. 2. **Definition:** Alternate interior angles ar

1. The problem asks to identify which angle is the alternate interior angle to angle 1 when three parallel lines are cut by a transversal. 2. Alternate interior angles are pairs of

1. **Problem statement:** An equilateral triangle is inscribed in a circle of radius 10 cm. Find the area of the triangle. 2. **Formula and important rules:**

1. The problem is to understand and simplify the expression $$D^2 = (4k)^2 + (3k)^2$$ and explain where the variable $k$ comes from. 2. This expression looks like the Pythagorean t

1. **State the problem:** We have triangle SWU with side SW = 45. Inside it, segment UV is parallel to SW and proportional. Given segments WV = 16 and TV = 15, we need to find the

1. **State the problem:** We have two similar triangles sharing vertex S. We know side lengths 45, 15, and 16, and we need to find the missing side length $UV$ in the smaller trian

1. **State the problem:** Find the length and midpoint of the line segment with endpoints $(-2, 6)$ and $(4, 0)$. 2. **Length formula:** The length $d$ between two points $(x_1, y_

1. **State the problem:** We have a polygon with an area of 129 cm² and some given side lengths, including an unknown length $x$. We need to find $x$. 2. **Analyze the shape:** The

1. **Problem statement:** Graph a circle with center at $(1,0)$ that passes through $(-3,0)$. Find the area and circumference of the circle, both in terms of $\pi$ and to the neare

1. The problem: Construct a triangle given the base length and the two adjacent angles. 2. Given: Base $AB = 8$ cm, angle $A = 50^\circ$, angle $B = 60^\circ$.

1. **Problem statement:** Prove that $\hat{A}BD = \hat{A}CE$ using the given figure with triangle $ABC$ and points $D$ and $E$ on the line through $B$ and $C$. 2. **Key idea:** Ang

1. The problem is to understand why the aspect ratio is 4:3. 2. Aspect ratio is the ratio of width to height of an image or screen.