{"id":298292,"date":"2022-09-10T21:06:17","date_gmt":"2022-09-10T15:36:17","guid":{"rendered":"https:\/\/infinitylearn.com\/surge\/question\/let-the-equations-of-two-sides-of-a-triangle-be-3x-2y60-and-4x-5y-20-0-if-the-orthocentre-of-this-triangle-is-at-11-then-the-equation-of-it\/"},"modified":"2022-09-10T21:06:17","modified_gmt":"2022-09-10T15:36:17","slug":"let-the-equations-of-two-sides-of-a-triangle-be-3x-2y60-and-4x-5y-20-0-if-the-orthocentre-of-this-triangle-is-at-11-then-the-equation-of-it","status":"publish","type":"questions","link":"https:\/\/infinitylearn.com\/surge\/question\/mathematics\/let-the-equations-of-two-sides-of-a-triangle-be3x-2y60an\/","title":{"rendered":"Let the equations of two sides of a triangle be\u00a03x-2y+6=0\u00a0and\u00a04x\u00a0+\u00a05y-\u00a020\u00a0=\u00a00. If the orthocentre of this triangle is at\u00a0(1,1)\u00a0then the equation of its third side is"},"content":{"rendered":"","protected":false},"author":1,"template":"","meta":{"_yoast_wpseo_focuskw":"","_yoast_wpseo_title":"","_yoast_wpseo_metadesc":"Let the equations of two sides of a triangle be\u00a03x-2y+6=0\u00a0and\u00a04x\u00a0+\u00a05y-\u00a020\u00a0=\u00a00. If the orthocentre of this triangle is at\u00a0(1,1)\u00a0then the equation of its third side is","custom_permalink":"question\/mathematics\/let-the-equations-of-two-sides-of-a-triangle-be3x-2y60an\/"},"categories":[],"acf":{"question":"

Let the equations of two sides of a triangle be 3<\/mn>x<\/mi>-<\/mo>2<\/mn>y<\/mi>+<\/mo>6<\/mn>=<\/mo>0<\/mn><\/math> and 4<\/mn>x<\/mi> <\/mo>+<\/mo> <\/mo>5<\/mn>y<\/mi>-<\/mo> <\/mo>20<\/mn> <\/mo>=<\/mo> <\/mo>0<\/mn><\/math>. If the orthocentre of this triangle is at (<\/mo>1<\/mn>,<\/mo>1<\/mn>)<\/mo><\/math> then the equation of its third side is<\/p>","options":[{"option":"

26x + 61y + 1675 = 0<\/p>","correct":false},{"option":"

122y + 26x + 1675 = 0<\/p>","correct":false},{"option":"

26x - 122y - 1675 = 0<\/p>","correct":true},{"option":"

122y - 26x - 1675 = 0<\/p>","correct":false}],"solution":"

Let the equation of side AB  <\/mtext>3<\/mn>x<\/mi>\u2212<\/mo>2<\/mn>y<\/mi>+<\/mo>6<\/mn>=<\/mo>0<\/mn><\/math>  <\/p>

                                                                                                          \u2026(i)<\/p>

and, the equation of side AC is 4x + 5y - 20 = 0<\/p>

                                                                                                          \u2026(ii)<\/p>

Solving (i) and (ii), we get vertex  A<\/mi>10<\/mn>23<\/mn><\/mfrac>,<\/mo>84<\/mn>23<\/mn><\/mfrac><\/mrow><\/mfenced><\/math>.<\/p>

Now, slope of A<\/mi>C<\/mi>=<\/mo>\u2212<\/mo>4<\/mn>\/<\/mo>5<\/mn><\/math><\/p>

\u2234<\/mo><\/math> Slope of perpendicular B<\/mi>E<\/mi>=<\/mo>5<\/mn>\/<\/mo>4<\/mn><\/math><\/p>

Also, BE passes through orthocentre H(l, 1)<\/p>

\u2234<\/mo><\/math> Equation of BE is<\/p>
<\/figure>

(<\/mo>y<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>=<\/mo>5<\/mn>4<\/mn><\/mfrac>(<\/mo>x<\/mi>\u2212<\/mo>1<\/mn>)<\/mo><\/math><\/p>

\u21d2<\/mo> <\/mspace>4<\/mn>y<\/mi>\u2212<\/mo>5<\/mn>x<\/mi>+<\/mo>1<\/mn>=<\/mo>0<\/mn><\/math>                                                      \u2026(iii)<\/p>

Solving (i) and (iii), we get vertex B<\/mi>\u2212<\/mo>13<\/mn>,<\/mo>\u2212<\/mo>33<\/mn>2<\/mn><\/mfrac><\/mrow><\/mfenced><\/math>.<\/p>

Now, slope of A<\/mi>B<\/mi>=<\/mo>3<\/mn>\/<\/mo>2<\/mn><\/math><\/p>

\u2234<\/mo><\/math> Slope of perpendicular C<\/mi>F<\/mi>=<\/mo>\u2212<\/mo>2<\/mn>\/<\/mo>3<\/mn><\/math><\/p>

Equation of CF is (<\/mo>y<\/mi>\u2212<\/mo>1<\/mn>)<\/mo>=<\/mo>\u2212<\/mo>2<\/mn><\/mrow>3<\/mn><\/mfrac>(<\/mo>x<\/mi>\u2212<\/mo>1<\/mn>)<\/mo><\/math><\/p>

\u21d2<\/mo>3<\/mn>y<\/mi>+<\/mo>2<\/mn>x<\/mi>\u2212<\/mo>5<\/mn>=<\/mo>0<\/mn><\/math>                                                          \u2026.(iv)                             <\/p>

Solving (ii) and (iv), we get vertex C<\/mi>35<\/mn>2<\/mn><\/mfrac>,<\/mo>\u2212<\/mo>10<\/mn><\/mrow><\/mfenced><\/math><\/p>

Equation of side BC is  (<\/mo>y<\/mi>+<\/mo>10<\/mn>)<\/mo>=<\/mo>\u2212<\/mo>10<\/mn>+<\/mo>33<\/mn>2<\/mn><\/mfrac><\/mrow>35<\/mn>2<\/mn><\/mfrac>+<\/mo>13<\/mn><\/mrow><\/mfrac>x<\/mi>\u2212<\/mo>35<\/mn>2<\/mn><\/mfrac><\/mrow><\/mfenced><\/math><\/p>

\u21d2<\/mo> <\/mspace>(<\/mo>y<\/mi>+<\/mo>10<\/mn>)<\/mo>=<\/mo>13<\/mn>61<\/mn><\/mfrac>x<\/mi>\u2212<\/mo>35<\/mn>2<\/mn><\/mfrac><\/mrow><\/mfenced>\u21d2<\/mo>26<\/mn>x<\/mi>\u2212<\/mo>122<\/mn>y<\/mi>=<\/mo>1675<\/mn><\/math><\/p>","subject":""},"yoast_head":"\nLet the equations of two sides of a triangle be\u00a03x-2y+6=0\u00a0and\u00a04x\u00a0+\u00a05y-\u00a020\u00a0=\u00a00. If the orthocentre of this triangle is at\u00a0(1,1)\u00a0then the equation of its third side is - Infinity Learn by Sri Chaitanya<\/title>\n<meta name=\"description\" content=\"Let the equations of two sides of a triangle be\u00a03x-2y+6=0\u00a0and\u00a04x\u00a0+\u00a05y-\u00a020\u00a0=\u00a00. 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