Triangles

To solve the problem, we need to find the measure of given that and . The figure shows a circle with center , and points , , and on the circumference.

Step 1: Understand the Geometry
is the center of the circle.
and are angles formed by radii and chords.
We need to find , which is the central angle subtended by the arc .
Step 2: Use the Inscribed Angle Theorem The key property here is that the central angle is twice the inscribed angle subtended by the same arc. However, we first need to find the measure of the inscribed angle .
Step 3: Find In :
(radii of the circle).
Therefore, is isosceles, and the base angles are equal.
Given , the other base angle is also .
The sum of angles in a triangle is : Similarly, in :
(radii of the circle).
Therefore, is isosceles, and the base angles are equal.
Given , the other base angle is also .
The sum of angles in a triangle is :
Step 4: Find The central angle is the sum of and : However, since is the reflex angle, the actual central angle we are interested in is:

Final Answer:

To solve the problem, we need to find the area of the triangle formed by the line with the coordinate axes.

1. Finding the Intercepts:
To find the x-intercept, set :

To find the y-intercept, set :

2. Using the Formula for Area of Triangle:
The area of a triangle formed by a line intersecting the coordinate axes at and is:

Substituting the intercepts:

Final Answer:
The area of the triangle is (Option A).

To solve the problem, we need to determine the measure of in , given that , , and .

1. Understanding Similar Triangles:
Since , the corresponding angles of the two triangles are equal. This means: Given and , we know:

2. Sum of Angles in a Triangle:
The sum of the interior angles in any triangle is . For , we have: Substituting the known values and :

3. Solving for :
Combine the known angle measures: Thus: Solving for :

Final Answer:
The measure of is .

To solve the problem, we need to determine the nature of the angle in the major segment of a circle. Let us analyze this step by step.

1. Understanding the Major Segment:
A major segment is the larger region of a circle that is bounded by a chord and the arc subtended by that chord. The key property here is the relationship between the angle subtended by the chord at the center of the circle and the angle subtended by the same chord at any point on the circumference of the circle.

2. Key Property of Angles in a Circle:
The angle subtended by a chord at the center of the circle is twice the angle subtended by the same chord at any point on the circumference. Mathematically, if is the angle subtended at the center and is the angle subtended at the circumference, then:

$ $

In the context of the major segment, the angle subtended at the center ( ) is greater than because the major segment is defined as the larger part of the circle. Therefore, the angle subtended at the circumference ( ) must be greater than .

3. Nature of the Angle in the Major Segment:
An angle greater than but less than is classified as an obtuse angle. Hence, the angle in the major segment is always obtuse.

Final Answer:
The correct option is .

To solve the problem, we need to determine the ratio of the area of to the area of , where , , and are the midpoints of the sides , , and , respectively.

1. Understanding the Midpoints and Medians:
When , , and are the midpoints of the sides of , the segments , , and are medians of the triangle. The medians of a triangle intersect at the centroid, which divides each median into a ratio of 2:1.

2. Properties of the Median Triangle:
The triangle formed by connecting the midpoints of the sides of a triangle (i.e., ) is called the **median triangle** or **midpoint triangle**. A key property of this triangle is that its area is exactly one-fourth of the area of the original triangle ( ). This can be proven using similarity and area ratios.

3. Similarity of Triangles:
Since , , and are midpoints, the sides of are parallel to and half the length of the corresponding sides of . Therefore, is similar to with a similarity ratio of .

The ratio of the areas of two similar triangles is the square of the ratio of their corresponding side lengths. Since the similarity ratio is , the area ratio is:

This means the area of is of the area of .

4. Final Answer:
The ratio of the area of to the area of is .

Final Answer:
The ratio is .

To solve the problem, we need to determine the length of in the given right triangle , where , , , and .

1. Understanding the Geometry:
Since , is the altitude from to the hypotenuse . The point divides into two segments: and . Therefore, the total length of is:

2. Using the Geometric Mean Theorem (Altitude-on-Hypotenuse Theorem):
The altitude to the hypotenuse of a right triangle creates two smaller right triangles that are similar to the original triangle and to each other. According to the geometric mean theorem: Substituting the given values: Thus:

3. Applying the Pythagorean Theorem in :
In , . Using the Pythagorean theorem: Substituting the known values: Thus:

Final Answer:
The length of is .

To solve the problem, we need to find the third vertex of a triangle given the centroid and two of its vertices.

1. Centroid Formula:
The centroid of a triangle with vertices , , and is given by:

2. Substituting Known Values:
Given:
Centroid
Vertex A = , Vertex B =
Let the third vertex be

Using the centroid formula:

3. Solving for x:

4. Solving for y:

5. Coordinates of the Third Vertex:
The third vertex is

Final Answer:
The third vertex is (3, -6).

To solve the problem, we need to find the length of side in triangle using the fact that .

1. Understanding Similar Triangles:

When two triangles are similar, the ratio of their areas is equal to the square of the ratio of their corresponding sides:

2. Substituting Known Values:

Given:

Area of

Area of

So,

3. Simplifying the Ratio:

Taking square roots on both sides:

4. Solving for :

Final Answer:

To solve the problem, we are given that , indicating that triangles and are similar by AA similarity.

1. Apply Triangle Similarity:
Since , the corresponding sides are proportional:

2. Plug in the Known Values:
Given:

3. Solve for DE:

Final Answer:
The length of is .

To solve the problem, we need to evaluate the expression in any triangle .

1. Understanding Angle Sum in a Triangle:
In any triangle, the sum of the interior angles is always:

2. Rearranging to Express :
From the above identity, we get:

3. Substituting into the Original Expression:
We are asked to find:

4. Applying Trigonometric Identity:
We know that:


Therefore,

Final Answer:
The value of is

To solve the problem, we need to find the length of segment using the concept of similar triangles formed by a line parallel to one side of a triangle.

1. Understanding the Geometry:
In the given triangle, , so triangles and are similar by Basic Proportionality Theorem (Thales' Theorem).

2. Given Information:

3. Use of Similar Triangles:
Since , we write the similarity condition:

4. Substituting the Known Values:

5. Cross Multiplying to Solve for :

Final Answer:
The length of is .

To solve the problem, we need to find the ratio of areas of two similar triangles when the ratio of their corresponding sides is given.

1. Understanding the Area Ratio for Similar Triangles:
If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of their corresponding sides.

Let the ratio of sides be , then:

2. Squaring the Side Ratio:

Final Answer:
The ratio of the areas of the triangles is