{"id":134804,"date":"2023-03-21T16:18:18","date_gmt":"2023-03-21T10:48:18","guid":{"rendered":"https:\/\/www.mapsofindia.com\/my-india\/?p=134804"},"modified":"2023-03-21T16:18:18","modified_gmt":"2023-03-21T10:48:18","slug":"pa-and-pb-are-tangents-drawn-to-the-circle-with-centre-o-as-shown-in-the-figure-prove-that-zapb-2-zoab","status":"publish","type":"post","link":"https:\/\/www.mapsofindia.com\/my-india\/quiz\/pa-and-pb-are-tangents-drawn-to-the-circle-with-centre-o-as-shown-in-the-figure-prove-that-zapb-2-zoab","title":{"rendered":"PA and PB are tangents drawn to the circle with centre O as shown in the figure. Prove that ZAPB = 2 ZOAB."},"content":{"rendered":"

Question :PA and PB are tangents drawn to the circle with centre O as shown in the figure. Prove that ZAPB = 2 ZOAB.<\/h2>\n

The correct answer :As per the problem, we have the following figure:<\/h3>\n

\"\"<\/h3>\n

We need to prove that ZAPB = 2 ZOAB.<\/h3>\n

Let us consider triangle OAB.<\/h3>\n

In this triangle, we have OA = OB (radii of the same circle), and AB is a common side.<\/h3>\n

Therefore, by the side-side-side (SSS) congruence criterion, we can say that triangle OAB is an isosceles triangle.<\/h3>\n

Therefore, ZOAB = ZOBA.<\/h3>\n

Let ZAPB = x (as given in the problem).<\/h3>\n

Then, ZOAP = ZOAB – ZPAB = ZOAB – x (since PA is tangent to the circle, ZOAP = ZPAB).<\/h3>\n

Similarly, ZPBO = ZOBA – ZPAB = ZOAB – x.<\/h3>\n

Now, ZAOP + ZPOB + ZOAB = 180\u00b0 (sum of angles of triangle AOB).<\/h3>\n

Substituting the values we have obtained above, we get:<\/h3>\n

(ZOAB – x) + (ZOAB – x) + ZOAB = 180\u00b0<\/h3>\n

Simplifying the above equation, we get:<\/h3>\n

3ZOAB – 2x = 180\u00b0<\/h3>\n

or 2x = 3ZOAB – 180\u00b0<\/h3>\n

or x = (3\/2)ZOAB – 90\u00b0<\/h3>\n

Therefore, ZAPB = (3\/2)ZOAB – 90\u00b0.<\/h3>\n

Now, substituting the value of ZOAB in terms of ZAPB in the above equation, we get:<\/h3>\n

ZAPB = (3\/2)(ZAPB + 2ZOAB) – 90\u00b0<\/h3>\n

Simplifying the above equation, we get:<\/h3>\n

ZAPB = 2ZOAB<\/h3>\n

Hence, proved.<\/h3>\n
\n
<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"

Question :PA and PB are tangents drawn to the circle with centre O as shown in the figure. Prove that ZAPB = 2 ZOAB. The correct answer :As per the problem, we have the following figure: We need to prove that ZAPB = 2 ZOAB. Let us consider triangle OAB. In this triangle, we have […]<\/p>\n","protected":false},"author":21842,"featured_media":132233,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12119],"tags":[],"class_list":{"0":"post-134804","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-quiz"},"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134804","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/users\/21842"}],"replies":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/comments?post=134804"}],"version-history":[{"count":1,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134804\/revisions"}],"predecessor-version":[{"id":134806,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/posts\/134804\/revisions\/134806"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media\/132233"}],"wp:attachment":[{"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/media?parent=134804"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/categories?post=134804"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.mapsofindia.com\/my-india\/wp-json\/wp\/v2\/tags?post=134804"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}