Anshu cuts a paper triangle A B C . He folds the paper perpendicular to B C such that it passes through the vertex A . He marks the point where the fold crosses B C as M . He unfolds the paper. He again folds all the three corners such that vertices A , B and C touch M without overlapping. Radhika performs a similar activity. She marks M by folding ∆ A B C such that it halves the side B C . When she folds the three corners such that vertices A , B and C touch M , unlike Anshu, the paper corners overlap. What can be the reason for overlapping?

Given, An similar activity performed by two different people. Anshu's activity shows no overlapping but Radhika's activity shows overlapping. We need to find the reason for overlapping. We know that, The median of a triangle is the line segment joining a vertex to the mid-point of the other side of a triangle. A triangle contains three medians, one from each vertex. The altitude of a triangle is perpendicular from a vertex to the opposite side of a triangle is called an altitude. A triangle contains three altitudes, one from each vertex. In case Anshu's case, he marked the altitude of the side B C , while Radhika marked the median of the side B C . A M is an altitude from the vertex A to its opposite side B C . Hence, the required answer is shown above.

In the figure shown below, P Q R is a straight line. The measure of ∠ P X Q = 20 ° . What is the measure of ∠ P X R ?

Given, In the figure shown below, P Q R is a straight line. The measure of ∠ P X Q = 20 ° . We know that, The angles opposite to the two equal sides of the triangle is always equal. A right isosceles triangle has a third angle as 90 ° . In △ P X Q , ∠ P X Q + ∠ P Q X + ∠ X P Q = 180 ° 20 ° + x + x = 180 ° 20 ° + 2 x = 180 ° 2 x = 180 ° - 20 ° 2 x = 160 ° x = 80 ° Thus, ∠ X P Q and ∠ P Q X = 80 ° . We know that, A straight line has 180 ° . P Q R is a straight line. ∠ X Q P + ∠ X Q R = 180 ° 180 ° + ∠ X Q R = 180 ° ∠ X Q R = 180 ° - 80 ° ∠ X Q R = 100 ° Thus, the measure of ∠ X Q R is 180 ° . △ X R Q is a isosceles triangle that has two sides are equal. ∠ X Q R + ∠ Q X R + ∠ X R Q = 180 ° 100 ° + ∠ Q X R + ∠ X R Q = 180 ° 100 ° + x + x = 180 ° 100 ° + 2 x = 180 ° 2 x = 180 ° - 100 ° 2 x = 80 ° x = 40 ° ∠ P X R = 20 ° + 40 ° ∠ P X R = 60 ° Hence, the measure of ∠ P X R is 60 ° .

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In an equilateral triangle $A B C$ , the length of $A C = 10 cm$ and altitude $A D = 6 cm$ . $P$ is a point on $A B$ .

The length of $B P = 4 x - 1$ . The length of $P A = 3 x + 4$ .

What is the length of $B P$ ?

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Important Questions on The Triangle and its Properties

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7th CBSE

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Mathematics Textbook of Competency Based Questions for Class VII>Chapter 6 - The Triangle and its Properties>Exercise>Q 2

In an equilateral triangle $A B C$ , the length of $A C = 10 cm$ and altitude $A D = 6 cm$ . $P$ is a point on $A B$ .

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The length of $B P = 4 x - 1$ . The length of $P A = 3 x + 4$ .

What is the length of the median on $B C$ ?

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7th CBSE

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Mathematics Textbook of Competency Based Questions for Class VII>Chapter 6 - The Triangle and its Properties>Exercise>Q 3

Anshu cuts a paper triangle $A B C$ . He folds the paper perpendicular to $B C$ such that it passes through the vertex $A$ . He marks the point where the fold crosses $B C$ as $M$ . He unfolds the paper. He again folds all the three corners such that vertices $A, B$ and $C$ touch $M$ without overlapping.

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Radhika performs a similar activity. She marks $M$ by folding $∆ A B C$ such that it halves the side $B C$ . When she folds the three corners such that vertices $A, B$ and $C$ touch $M$ , unlike Anshu, the paper corners overlap.