In triangle ABC, AD is perpendicular to side BC and AD 2 = BD × DC. Show that angle BAC = 90°.

Triangle ABC is shown in the figure below: Given : AD 2 = BD × DC ⇒ ∠ADB = ∠ADC [Both = 90°] ∴ △DBA ~ △DAC (By SAS). Since, triangles are similar they will be equiangular. ∴ ∠1 = ∠C and ∠2 = ∠B ⇒ ∠1 + ∠2 = ∠B + ∠C ⇒ ∠A = ∠B + ∠C By angle sum property : ⇒ ∠A + ∠B + ∠C = 180° ⇒ ∠A + ∠A = 180° ⇒ 2∠A = 180° ⇒ ∠A = 90°. From figure, ⇒ ∠BAC = ∠A = 90°. Hence, proved that ∠BAC = 90°.

In △ABC, ∠B = 90° and BD ⊥ AC.
(i) If CD = 10 cm and BD = 8 cm; find AD.
(ii) If AC = 18 cm and AD = 6 cm; find BD.
(iii) If AC = 9 cm and AB = 7 cm; find AD.

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In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm, L is a point on PR such that RL : LP = 2 : 3. QL produced meets RS at M and PS produced at N. Find the lengths of PN and RM.

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In the given figure, AB || EF || DC; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.
(i) Name the three pairs of similar triangles.
(ii) Find the length of EC and EF.

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In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.
(i) Calculate the ratio PQ : AC, giving reason for your answer.
(ii) In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.

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Mathematics

In triangle ABC, AD is perpendicular to side BC and AD² = BD × DC.
Show that angle BAC = 90°.

Similarity

Related Questions

In △ABC, ∠B = 90° and BD ⊥ AC.
(i) If CD = 10 cm and BD = 8 cm; find AD.
(ii) If AC = 18 cm and AD = 6 cm; find BD.
(iii) If AC = 9 cm and AB = 7 cm; find AD.

In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm, L is a point on PR such that RL : LP = 2 : 3. QL produced meets RS at M and PS produced at N. Find the lengths of PN and RM.

In the given figure, AB || EF || DC; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.
(i) Name the three pairs of similar triangles.
(ii) Find the length of EC and EF.

In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.
(i) Calculate the ratio PQ : AC, giving reason for your answer.
(ii) In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.

Mathematics

In triangle ABC, AD is perpendicular to side BC and AD2 = BD × DC.Show that angle BAC = 90°.

Similarity

Related Questions

In △ABC, ∠B = 90° and BD ⊥ AC.(i) If CD = 10 cm and BD = 8 cm; find AD.(ii) If AC = 18 cm and AD = 6 cm; find BD.(iii) If AC = 9 cm and AB = 7 cm; find AD.

In the figure, PQRS is a parallelogram with PQ = 16 cm and QR = 10 cm, L is a point on PR such that RL : LP = 2 : 3. QL produced meets RS at M and PS produced at N. Find the lengths of PN and RM.

In the given figure, AB || EF || DC; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.(i) Name the three pairs of similar triangles.(ii) Find the length of EC and EF.

In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.(i) Calculate the ratio PQ : AC, giving reason for your answer.(ii) In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.

In triangle ABC, AD is perpendicular to side BC and AD² = BD × DC.
Show that angle BAC = 90°.

In △ABC, ∠B = 90° and BD ⊥ AC.
(i) If CD = 10 cm and BD = 8 cm; find AD.
(ii) If AC = 18 cm and AD = 6 cm; find BD.
(iii) If AC = 9 cm and AB = 7 cm; find AD.

In the given figure, AB || EF || DC; AB = 67.5 cm, DC = 40.5 cm and AE = 52.5 cm.
(i) Name the three pairs of similar triangles.
(ii) Find the length of EC and EF.

In the given figure, P is a point on AB such that AP : PB = 4 : 3. PQ is parallel to AC.
(i) Calculate the ratio PQ : AC, giving reason for your answer.
(ii) In triangle ARC, ∠ARC = 90° and in triangle PQS, ∠PSQ = 90°. Given QS = 6 cm, calculate the length of AR.