P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

Given: In Δ ABC, ∠C = 90° and P is mid point of AB. To prove: AB = 2 CP Construction: Draw line segment PK parallel to BC, which meets AC at point K. Proof: Since, PK ∥ BC and AC is traversal, ⇒ ∠AKP = ∠ACB = 90° = ∠PKC Also, as P is mid-point of AB and KP is parallel to CB; PK bisects side AC i.e., AK = KC. Now, in Δ APK and Δ CPK, KP = KP (Common) ∠PKA = ∠PKC (both are 90°) AK = KC (Since, KP ∥ BC and P is mid point of AB) By SAS congruency criterion, Δ APK ≅ Δ CPK By corresponding parts of congruent triangles, PA = PC As we know that P is mid point of AB. ⇒ PA = AB So, PA = PC = AB Hence, AB = 2 CP.

In the given figure, AB > AC and D is any point on BC.
Prove that : AB > AD.

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In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

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In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

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Mathematics

P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

Mid-point Theorem

Related Questions

In the given figure, AB > AC and D is any point on BC.
Prove that : AB > AD.

In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.

Mathematics

P is the mid-point of the hypotenuse AB of the right-angled triangle ABC. Prove that : AB = 2 CP.

Mid-point Theorem

Related Questions

In the given figure, AB > AC and D is any point on BC.Prove that : AB > AD.

In quadrilateral ABCD, side DC is largest. Show that AB + AD > DC - BC.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :(i) AD is perpendicular to FE.(ii) AD and FE bisect each other.

In the given figure, AB > AC and D is any point on BC.
Prove that : AB > AD.

In △ABC, AB = AC. D, E and F are mid-points of the sides BC, CA and AB respectively. Show that :
(i) AD is perpendicular to FE.
(ii) AD and FE bisect each other.