A median in a triangle is a line segment that extends from a vertex to the midpoint of the opposite side, dividing it into two equal parts.
Triangle Median Practice Problems and Worksheets
Master triangle medians with step-by-step practice problems. Learn to find medians, calculate areas, and solve centroid problems in triangles.
📚Master Triangle Medians with Interactive Practice
- Identify and draw all three medians from vertices to opposite side midpoints
- Calculate median lengths in right triangles using the hypotenuse rule
- Find areas of triangles created when medians divide the original triangle
- Solve problems involving medians in equilateral and isosceles triangles
- Locate the centroid where all three medians intersect
- Apply median properties to find missing side lengths and angles
Understanding Parts of a Triangle
Complete explanation with examples
Median in a triangle
Additional properties:
- In every triangle, it is possible to draw 3 medians.
- All 3 medians intersect at one point.
- The median to a side in a triangle creates 2 triangles of equal area.
- In an equilateral triangle - the median is also a height and an angle bisector.
- In an isosceles triangle - the median from the vertex angle is also a height and an angle bisector.
- In a right triangle - the median to the hypotenuse equals half the hypotenuse.
Practice Parts of a Triangle
Test your knowledge with 36 quizzes
DB is a side in triangle ABC
Examples with solutions for Parts of a Triangle
Step-by-step solutions included
Exercise #1
Can a triangle have a right angle?
Step-by-Step Solution
To determine if a triangle can have a right angle, consider the following explanation:
- Definition of a Right Angle: An angle is classified as a right angle if it measures exactly .
- Definition of a Right Triangle: A right triangle is a type of triangle that contains exactly one right angle.
- According to the definition, a right triangle specifically includes a right angle. This is a well-established classification of triangles in geometry.
Thus, a triangle can indeed have a right angle and is referred to as a right triangle.
Therefore, the solution to the problem is Yes.
Answer:
Yes
Video Solution
Exercise #2
Is the straight line in the figure the height of the triangle?
Step-by-Step Solution
The triangle's altitude is a line drawn from a vertex perpendicular to the opposite side. The vertical line in the diagram extends from the triangle's top vertex straight down to its base. By definition of altitude, this line is the height if it forms a right angle with the base.
To solve this problem, we'll verify that the line in question satisfies the altitude condition:
- Step 1: Identify the triangle's vertices and base. From the diagram, the base appears horizontal, and the vertex lies directly above it.
- Step 2: Check the nature of the line. The line is vertical when the base is horizontal, indicating perpendicularity.
- Conclusion: The vertical line forms right angles with the base, thus acting as the altitude or height.
Therefore, the straight line depicted is indeed the height of the triangle. The answer is Yes.
Answer:
Yes
Video Solution
Exercise #3
Is the straight line in the figure the height of the triangle?
Step-by-Step Solution
To determine if the straight line is the height of the triangle, we'll analyze its role within the triangle:
- Step 1: Observe the triangle and the given line. The triangle seems to be made of three sides and a vertical line within it.
- Step 2: Recall that the height of a triangle, in geometry, is defined as a perpendicular dropped from a vertex to the opposite side.
- Step 3: Examine the positioning of the line: The vertical line starts at one vertex of the triangle and intersects the base, appearing to be perpendicular.
- Step 4: Verify perpendicularity: Given that the line is shown as a clear vertical (and a small perpendicular indicator suggests perpendicularity), we accept this line as the height.
- Step 5: Conclude with verification that the line is effectively meeting the definition of height for the triangle in the diagram.
Therefore, the vertical line in the figure is indeed the height of the triangle.
Yes
Answer:
Yes
Video Solution
Exercise #4
Can a plane angle be found in a triangle?
Step-by-Step Solution
To determine whether a plane angle can be found in a triangle, we need to understand what a plane angle is and compare it to the angles within a triangle.
- A plane angle is an angle formed by two lines lying in the same plane.
- In the context of geometry, angles found within a triangle are the interior angles, which are the angles between the sides of the triangle.
- Although the angles in a triangle are indeed contained within a plane (since a triangle itself is a planar figure), when referencing "plane angles" in geometry, we usually consider angles related to different geometric configurations beyond those specifically internal to defined planar shapes like a triangle.
- The term "plane angle" typically refers to the measurement of an angle in radians or degrees within a plane, but this doesn't specifically pertain to angles of a triangle.
Therefore, based on the context and usual geometric conventions, the concept of a "plane angle" is not typically used to describe the angles found within a triangle. Thus, a plane angle as defined generally in geometry is not found specifically within a triangle.
Therefore, the correct answer to the question is .
Answer:
No
Video Solution
Exercise #5
Is the straight line in the figure the height of the triangle?
Step-by-Step Solution
To determine if the given straight line is the height of the triangle, we need to check whether it is perpendicular to the side it intersects (or its extension), which is the definition of a height (altitude) in a triangle.
Looking at the figure, the straight line is drawn from a vertex of the triangle to a point on the opposite side, but it is not perpendicular to that side or its extension. Therefore, the line does not meet the criteria for being a height of the triangle.
In conclusion, the line is not the height of the triangle because it is not perpendicular to the opposite side.
Therefore, the correct answer to the problem is No.
Answer:
No
Video Solution
Frequently Asked Questions
How do you find the median of a triangle step by step?
+To find a triangle median: 1) Identify a vertex and the opposite side, 2) Find the midpoint of the opposite side by dividing its length by 2, 3) Draw a line segment from the vertex to this midpoint. This line segment is the median, and it divides the opposite side into two equal parts.
What is the difference between median, altitude, and angle bisector in triangles?
+A median connects a vertex to the midpoint of the opposite side. An altitude is perpendicular from a vertex to the opposite side. An angle bisector divides an angle into two equal parts. In equilateral triangles, all three are the same line, while in isosceles triangles, they coincide only for the vertex angle.
How many medians does every triangle have and where do they meet?
+Every triangle has exactly 3 medians - one from each vertex to the midpoint of the opposite side. All three medians always intersect at a single point called the centroid, which divides each median in a 2:1 ratio from vertex to midpoint.
What is the median to hypotenuse rule in right triangles?
+In a right triangle, the median drawn to the hypotenuse equals half the length of the hypotenuse. If the hypotenuse is 10 units, then the median to the hypotenuse is 5 units. This creates two equal segments from the right angle vertex to each end of the hypotenuse.
How do you calculate the area when a median divides a triangle?
+When a median divides a triangle, it creates two smaller triangles with equal areas. Each smaller triangle has an area equal to half the original triangle's area. Use the formula: Area = (base × height) ÷ 2, where the base is half the original side length.
What are the special properties of medians in isosceles triangles?
+In an isosceles triangle, the median drawn from the vertex angle (between the two equal sides) has special properties: it is also the altitude (height) and the angle bisector. This median is perpendicular to the base and divides the vertex angle into two equal parts.
How do you solve triangle median problems involving multiple medians?
+For problems with multiple medians: 1) Use the fact that medians bisect sides to find segment lengths, 2) Apply the equal area property - medians create triangles with equal areas, 3) Remember that all medians meet at the centroid, 4) Use specific triangle properties (right, isosceles, equilateral) when applicable.
What grade level typically learns about triangle medians in geometry?
+Triangle medians are typically introduced in middle school geometry (grades 7-8) and studied more extensively in high school geometry (grades 9-10). Students learn basic median properties first, then progress to more complex applications involving centroids and area calculations.