{"id":98,"date":"2020-02-15T04:40:52","date_gmt":"2020-02-15T04:40:52","guid":{"rendered":"http:\/\/mathfun4kids.com\/mlog\/?p=98"},"modified":"2020-11-10T02:29:34","modified_gmt":"2020-11-10T02:29:34","slug":"mathcount-exercises-21","status":"publish","type":"post","link":"https:\/\/mathfun4kids.com\/mlog\/?p=98","title":{"rendered":"MATHCOUNTS Exercises – 21"},"content":{"rendered":"\n

The perimeter of rhombus $ABCD$ is 8 and its area is 2. Find the value of $\\angle{ABC}$ in terms of $\\pi$. Click here for the solution.<\/a><\/p>\n

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Solution<\/strong> Draw line $AE$ perpendicular to $BC$ with $E$ on $BC$. Connect $A$ and $C$ to form two congruent $\\triangle{ABC}$ and $\\triangle{ADC}$. Because $ABCD$ is a rhombus, we have\n$$ AB=BC=CD=DA=\\frac{1}{4} \\cdot \\text{perimeter of ABCD} =\\frac{1}{4}\\cdot 8=2$$ $$[\\triangle{ABC}]=[\\triangle{BCD}]=\\frac{1}{2}\\cdot[ABCD]=\\frac{1}{2}\\cdot 2\\sqrt{2}=\\sqrt{2}$$\n$$ \\because [\\triangle{ABC}]=\\frac{1}{2}\\cdot BC \\cdot AE $$\n$$ \\therefore AE=2\\cdot\\frac{[\\triangle{ABC}]}{BC}=2\\cdot\\frac{\\sqrt{2}}{2}=\\sqrt{2} $$\n$$ $$\n$$ \\because \\triangle{ABE}\\ \\text{is a right triangle} $$\n$$ \\therefore BE = \\sqrt{AB^2-AE^2}=\\sqrt{2^2-(\\sqrt{2})^2}=\\sqrt{2} $$\n$$ \\because AE = BE, \\triangle{ABC}\\ \\text{is an isosceles right triangle} $$\n$$ \\therefore \\angle{ABC}=45^{\\circ}=\\frac{\\pi}{4} $$\n<\/p>\n<\/div>\n\n\n\n

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The perimeter of rhombus $ABCD$ is 8 and its area is 2. Find the value of $\\angle{ABC}$ in terms of $\\pi$. Click here for the solution. Solution Draw line $AE$ perpendicular to $BC$ with $E$ on $BC$. Connect $A$ and $C$ to form two congruent $\\triangle{ABC}$ and … Continue reading →<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false},"categories":[9,11],"tags":[],"_links":{"self":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/98"}],"collection":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=98"}],"version-history":[{"count":16,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/98\/revisions"}],"predecessor-version":[{"id":953,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=\/wp\/v2\/posts\/98\/revisions\/953"}],"wp:attachment":[{"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=98"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=98"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mathfun4kids.com\/mlog\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=98"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}