📐 geometry

1. **Problem statement:** We are given a triangle with an incenter and need to find the measures of angles and segments related to the incenter and the triangle. 2. **Key concept:*

1. **Problem:** Apply the translation $T_{(3, -1)}$ to point $E(1, 5)$. 2. **Translation formula:** For a point $(x, y)$ and translation $T_{(a, b)}$, the image point is $(x', y')

1. The problem asks to determine if the statement "Two segments that have the same length must be congruent" is sometimes, always, or never true. 2. The definition of congruent seg

1. **State the problem:** We need to find the measure of angle $B$ (denoted as $m\angle B$) in triangle $ABC$ where the angles are given as follows: - Angle $A = 60^\circ$

1. **State the problem:** We need to find the measure of angle $\angle BCD$ in the given triangle and external angle setup. 2. **Identify known angles and variables:**

1. **State the problem:** We need to find the area of the top of a plant stand, which is a regular octagon with side length $2.7$ inches. 2. **Formula and explanation:** A regular

1. **State the problem:** Amy has an original image with dimensions width $= \frac{2}{3}$ ft and height $= \frac{1}{2}$ ft.

1. **State the problem:** We have two triangles, Figure A and Figure B, where Figure B is a dilation of Figure A centered at the origin. We need to find the scale factor of this di

1. **Problem Statement:** We have a dilation centered at the origin that transforms Figure A into Figure B. We need to find the scale factor of this dilation.

1. **State the problem:** We need to find the unknown interior angle $x$ of a pentagon given the other four interior angles: $84^\circ$, $153^\circ$, $131^\circ$, and $105^\circ$.

1. **State the problem:** We have a triangle with interior angles labeled as $12a$, $4a$, and $36^\circ$. We need to find the value of $a$. 2. **Recall the rule for interior angles

1. The problem asks to find the supplement of an angle measuring 37°39'17". 2. The supplement of an angle is found using the formula: $$\text{Supplement} = 180^\circ - \text{angle}

1. The problem involves identifying and understanding the subtraction expressions with polygon shapes in a 5x3 grid. 2. Each cell contains a subtraction sign followed by a polygon

1. **Problem:** The scale of a map is 1 : 250000. On the map, the distance between two towns is 8.2 cm. Work out the real-life distance between the two towns, giving your answer in

1. **Problem:** The scale of a map is 1 : 250000. On the map, the distance between two towns is 8.2 cm. Work out the real-life distance between the two towns, giving your answer in

1. **State the problem:** We have a polygon VWXY with sides VW = XY, WX = YV, and all interior angles equal to 90°. 2. **Identify the polygon type:** Since all angles are 90°, VWXY

1. **State the problem:** We are given a polygon with sides 16, 32, 32, and 16 units, opposite sides equal and parallel, and pairs of opposite angles equal. We need to identify whi

1. **State the problem:** Rotate the grey parallelogram with vertices at points $(0,1)$, $(2,3)$, $(3,2)$, and $(1,0)$ by 90° clockwise about the origin $(0,0)$. 2. **Formula for r

1. **Problem statement:** Rotate the triangle with vertices at $(-3,-5)$, $(-1,4)$, and $(3,4)$ by 90° anticlockwise about the point $(-1,1)$. 2. **Formula for rotation about a poi

1. **State the problem:** We need to find the measure of angle $\angle CAE$ given the angles $40^\circ$ and $104^\circ$ in a figure with two parallel lines and intersecting lines.

1. The problem asks to find all points that are the same distance from Point A as Point B is from Point A. 2. First, identify the coordinates of Point A and Point B.