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In right triangle ABC, right-angled at C, M is the midpoint of hypotenuse AB. C is joined to M and produced to a point D such that DM = CM. Point D is joined to point B. show that:1) ΔAMC ≅ ΔBMD2) ∠DBC is a right angle3) Δ DBC ≅ ΔACB4) CM = 1/2 AB |
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Answer» Step-by-step explanation: i) △AMC≅△BMD Proof: As 'M' is the midpoint BM=AM And also it is the mid point of DC then DM=MC And AC=DB (same LENGTH) ∴Therefore we can say that ∴△AMC≅△BMD ii) ∠DBC is a RIGHT angle As △DBC is a right angle triangle and DC ² =DB²+BC ² (Pythagoras) So, ∠B=90° ∴∠DBC is 90° iii) △DBC≅△ACB As M is the midpoint of AB and DC. So, DM=MC and AB=BM ∴DC=AB (As they are in same length) And also, AC=DB and ∠B=∠C=90° By SAS Axiom ∴△DBC≅△ACB iv) CM= 1 /2 AB As △DBC≅△ACB CM= DC/2 ∴DC=AB(△DBC≅△ACB) So, CM= AB/2 ∴CM= 1/2 AB |
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