A school designing a triangular garden △ABC. To construct a walking path inside the garden, the gardener marks the mid-points of two sides: point D is the mid-point of side AB and point E is the midpoint of side AC. The path DE is drawn to connect these mid-points. The length of side BC of the triangular garden is 12 m, AB = 10 m and AC = 10 m.
Based on the above information answer the following:
(i) What is the length of path DE?
(ii) Assign a special name to Quadrilateral BCED and find its perimeter.

Question

A school designing a triangular garden △ABC. To construct a walking path inside the garden, the gardener marks the mid-points of two sides: point D is the mid-point of side AB and point E is the midpoint of side AC. The path DE is drawn to connect these mid-points. The length of side BC of the triangular garden is 12 m, AB = 10 m and AC = 10 m.

Based on the above information answer the following:

(i) What is the length of path DE?

(ii) Assign a special name to Quadrilateral BCED and find its perimeter.

KnowledgeBoat · Accepted Answer

Given, AB = AC = 10 m BC = 12 m D and E are midpoints of AB and AC (i) By mid-point theorem, Line joining midpoints of two sides of a triangle is parallel to the third side and equal to half of it. So, DE = × BC = × 12 = 6 m ∴ DE = 6 m. Hence, length of DE = 6 m. (ii) Since D and E are mid-points, the Mid-point Theorem tells us that DE || BC. ∴ Only one pair of opposite sides are parallel. So, A quadrilateral with at least one pair of parallel opposite sides is called a Trapezium. Given, BC = 12 m and DE = 6 m Since D and E are midpoints, so by mid-point theorem. BD = AB = × 10 = 5 m CE = AC = × 10 = 5 m Perimeter = sum of all sides Perimeter of quadrilateral BCED = BC + CE + ED + DB = 12 + 5 + 6 + 5 = 28 m. Hence, quadrilateral BCED is a trapezium and its perimeter = 28 m.

Mathematics

Mid-point Theorem

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