Test: A Tale of Three Intersecting Lines - Class 7 MCQ

A triangle is defined by the following key components:

• Three sides: These are the straight edges that form the triangle.
• Three angles: The angles between each pair of sides.
• Three vertices: The points where the sides meet.

These components are essential for identifying and understanding the properties of a triangle.

The altitude of a triangle is defined as the perpendicular line drawn from a vertex to the opposite side. Here are some key points to understand about the altitude:

• The altitude can be located inside the triangle for acute triangles.
• In right triangles, one of the legs serves as the altitude.
• For obtuse triangles, the altitude extends outside the triangle.
• The length of the altitude is essential for calculating the area of the triangle.

The Triangle Inequality Theorem states that for any triangle, the sum of the lengths of any two sides must be greater than the length of the third side.

This theorem is fundamental in understanding the properties of triangles.

According to the triangle inequality theorem, a triangle can exist if the sum of the lengths of any two sides is greater than the length of the third side. Let's examine the sides:

• 3 cm + 4 cm > 6 cm (7 > 6) - True
• 3 cm + 6 cm > 4 cm (9 > 4) - True
• 4 cm + 6 cm > 3 cm (10 > 3) - True

Since all three conditions are satisfied, the triangle with sides of 3 cm, 4 cm, and 6 cm can exist.

 When one angle measures exactly 90 degrees, it is known as a Right-Angled Triangle. Here are some key points about this type of triangle:

• A right-angled triangle has one angle that is a right angle.
• The other two angles are acute, meaning they are less than 90 degrees.
• Right-angled triangles follow the Pythagorean theorem, which states that the square of the hypotenuse is equal to the sum of the squares of the other two sides.

A triangle cannot exist with side lengths of 2 cm, 3 cm, and 6 cm because:

• The sum of the two shorter sides (2 cm + 3 cm = 5 cm) is less than the longest side (6 cm).
• This situation violates the triangle inequality theorem.

Therefore, such a triangle cannot be formed.

In triangle ABC, the sum of all angles equals 180 degrees. To find the measure of angle C, follow these steps:

• Given angle A = 25 degrees
• Given angle B = 70 degrees
• Calculate angle C as follows:
• Angle C = 180 - (Angle A + Angle B)
• Angle C = 180 - (25 + 70)
• Angle C = 180 - 95
• Angle C = 85 degrees

Thus, the measure of angle C is 85 degrees.

An equilateral triangle is a type of triangle where all three sides are of equal length. In contrast, other triangle types include:

• Scalene Triangle: All sides and angles are different.
• Isosceles Triangle: Two sides are of equal length.
• Right-Angled Triangle: One angle measures 90 degrees.

A triangle with all angles measuring less than 90 degrees is classified as an acute-angled triangle.

In an isosceles triangle, the angles opposite the equal sides have a specific relationship:

• The two angles opposite the equal sides are always equal in measure.
• This property holds true regardless of whether the angles are acute or obtuse.

The corner points of a triangle are known as vertices. Each triangle has three vertices, which are the points where the sides meet.

• Midpoints are the centre points of the sides, not the corners.
• Sides refer to the edges of the triangle, not the corner points.
• Angles are formed by the sides at the vertices, but they are not the points themselves.

Thus, the correct term for the corner points is vertices.

An obtuse-angled triangle is defined by having one angle that is greater than 90 degrees. The other two angles must be acute, meaning they are each less than 90 degrees. This arrangement is necessary to comply with the angle sum property, which states that the total of all angles in a triangle is always 180 degrees.

Key features of obtuse-angled triangles include:

• One angle exceeding 90 degrees.
• Two angles that are acute.
• Applications in trigonometry for various calculations.

The term used to describe three points that lie on the same straight line is collinear.

• Collinear: Points that are on the same line.
• Parallel: Lines that never meet.
• Vertical: An orientation that is up and down.
• Tangent: A line that touches a curve at one point.

To construct a triangle with two sides and the included angle, follow these steps:

• Draw the first side to the required length.
• At one endpoint of this line, use a protractor to create the specified angle.
• From the angle's vertex, measure the length of the second side along the angle's arm and mark this point.
• Connect this marked point back to the other endpoint of the first side to complete the triangle.

This method effectively demonstrates the use of geometric principles in construction and design.

The point where two arcs intersect represents the third vertex of the triangle.

• Midpoint: This divides a line segment into two equal parts.
• Radius: This is the distance from the centre of a circle to its edge.
• Third vertex: This is the intersection point of the arcs, essential for forming the triangle.
• Angle bisector: This line divides an angle into two equal angles.