Math / Geometry

QuestionThe perimeter of a triangle is 75 cm . The longest side is 7 cm less than the sum of the other two sides. Twice the shortest side is 6 cm less than the longest side. Find the length of each side of the triangle.
The shortest side is $\square$ $\square$ The middle side is $\square$ $\square$ The longest side is $\square$ $\square$

Studdy Solution

STEP 1

1. Let the sides of the triangle be denoted as $a$ , $b$ , and $c$ where $a < b < c$ .
2. The perimeter of the triangle is given as 75 cm, so we have the equation $a + b + c = 75$ .
3. The longest side $c$ is 7 cm less than the sum of the other two sides, so $c = a + b - 7$ .
4. Twice the shortest side $a$ is 6 cm less than the longest side, so $2a = c - 6$ .

STEP 2

1. Express $c$ in terms of $a$ and $b$ using the given conditions.
2. Substitute this expression for $c$ into the perimeter equation.
3. Express $b$ in terms of $a$ using the equations derived.
4. Solve for $a$ .
5. Use the value of $a$ to find $b$ and $c$ .

STEP 3

Use the given condition that $c = a + b - 7$ to express $c$ in terms of $a$ and $b$ .
$c = a + b - 7$

STEP 4

Substitute $c = a + b - 7$ into the perimeter equation $a + b + c = 75$ .
$a + b + (a + b - 7) = 75$

STEP 5

Simplify the equation to combine like terms.
$2a + 2b - 7 = 75$

STEP 6

Solve for $2a + 2b$ .
$2a + 2b = 82$

STEP 7

Divide the equation by 2 to isolate $a + b$ .
$a + b = 41$

STEP 8

Use the given condition $2a = c - 6$ to express $c$ in terms of $a$ .
$c = 2a + 6$

STEP 9

Substitute the expression for $c$ from STEP_6 into the equation $c = a + b - 7$ .
$2a + 6 = a + b - 7$

STEP 10

Solve for $b$ .
$2a + 6 = a + b - 7$ $2a - a + 6 + 7 = b$ $a + 13 = b$

STEP 11

Substitute $b = a + 13$ into the equation $a + b = 41$ from STEP_5.
$a + (a + 13) = 41$ $2a + 13 = 41$

STEP 12

Solve for $a$ .
$2a = 28$ $a = 14$

STEP 13

Use $a = 14$ to find $b$ .
$b = a + 13$ $b = 14 + 13$ $b = 27$

STEP 14

Use $a = 14$ to find $c$ .
$c = 2a + 6$ $c = 2(14) + 6$ $c = 28 + 6$ $c = 34$
Solution: The shortest side is $14$ cm. The middle side is $27$ cm. The longest side is $34$ cm.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Express $c$ in terms of $a$ and $b$ using the given conditions.
2. Substitute this expression for $c$ into the perimeter equation.
3. Express $b$ in terms of $a$ using the equations derived.
4. Solve for $a$ .
5. Use the value of $a$ to find $b$ and $c$ .

STEP 3

Use the given condition that $c = a + b - 7$ to express $c$ in terms of $a$ and $b$ .
$c = a + b - 7$

STEP 4

Substitute $c = a + b - 7$ into the perimeter equation $a + b + c = 75$ .
$a + b + (a + b - 7) = 75$

STEP 5

Simplify the equation to combine like terms.
$2a + 2b - 7 = 75$

STEP 6

Solve for $2a + 2b$ .
$2a + 2b = 82$

STEP 7

Divide the equation by 2 to isolate $a + b$ .
$a + b = 41$

STEP 8

Use the given condition $2a = c - 6$ to express $c$ in terms of $a$ .
$c = 2a + 6$

STEP 9

Substitute the expression for $c$ from STEP_6 into the equation $c = a + b - 7$ .
$2a + 6 = a + b - 7$

STEP 10

Solve for $b$ .
$2a + 6 = a + b - 7$ $2a - a + 6 + 7 = b$ $a + 13 = b$

STEP 11

Substitute $b = a + 13$ into the equation $a + b = 41$ from STEP_5.
$a + (a + 13) = 41$ $2a + 13 = 41$

STEP 12

Solve for $a$ .
$2a = 28$ $a = 14$

STEP 13

Use $a = 14$ to find $b$ .
$b = a + 13$ $b = 14 + 13$ $b = 27$

STEP 14

Use $a = 14$ to find $c$ .
$c = 2a + 6$ $c = 2(14) + 6$ $c = 28 + 6$ $c = 34$
Solution: The shortest side is $14$ cm. The middle side is $27$ cm. The longest side is $34$ cm.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

1. Express ccc in terms of aaa and bbb using the given conditions.2. Substitute this expression for ccc into the perimeter equation.3. Express bbb in terms of aaa using the equations derived.4. Solve for aaa.5. Use the value of aaa to find bbb and ccc.

STEP 3

Use the given condition that c=a+b−7c = a + b - 7c=a+b−7 to express ccc in terms of aaa and bbb.c=a+b−7 c = a + b - 7 c=a+b−7

STEP 4

Substitute c=a+b−7c = a + b - 7c=a+b−7 into the perimeter equation a+b+c=75a + b + c = 75a+b+c=75.a+b+(a+b−7)=75 a + b + (a + b - 7) = 75 a+b+(a+b−7)=75

STEP 5

Simplify the equation to combine like terms.2a+2b−7=75 2a + 2b - 7 = 75 2a+2b−7=75

STEP 6

Solve for 2a+2b2a + 2b2a+2b.2a+2b=82 2a + 2b = 82 2a+2b=82

STEP 7

Divide the equation by 2 to isolate a+ba + ba+b.a+b=41 a + b = 41 a+b=41

STEP 8

Use the given condition 2a=c−62a = c - 62a=c−6 to express ccc in terms of aaa.c=2a+6 c = 2a + 6 c=2a+6

STEP 9

Substitute the expression for ccc from STEP_6 into the equation c=a+b−7c = a + b - 7c=a+b−7.2a+6=a+b−7 2a + 6 = a + b - 7 2a+6=a+b−7

STEP 10

Solve for bbb.2a+6=a+b−7 2a + 6 = a + b - 7 2a+6=a+b−7 2a−a+6+7=b 2a - a + 6 + 7 = b 2a−a+6+7=b a+13=b a + 13 = b a+13=b

STEP 11

Substitute b=a+13b = a + 13b=a+13 into the equation a+b=41a + b = 41a+b=41 from STEP_5.a+(a+13)=41 a + (a + 13) = 41 a+(a+13)=41 2a+13=41 2a + 13 = 41 2a+13=41

STEP 12

Solve for aaa.2a=28 2a = 28 2a=28 a=14 a = 14 a=14

STEP 13

Use a=14a = 14a=14 to find bbb.b=a+13 b = a + 13 b=a+13 b=14+13 b = 14 + 13 b=14+13 b=27 b = 27 b=27

STEP 14

Use a=14a = 14a=14 to find ccc.c=2a+6 c = 2a + 6 c=2a+6 c=2(14)+6 c = 2(14) + 6 c=2(14)+6 c=28+6 c = 28 + 6 c=28+6 c=34 c = 34 c=34Solution: The shortest side is 141414 cm. The middle side is 272727 cm. The longest side is 343434 cm.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Express $c$ in terms of $a$ and $b$ using the given conditions.
2. Substitute this expression for $c$ into the perimeter equation.
3. Express $b$ in terms of $a$ using the equations derived.
4. Solve for $a$ .
5. Use the value of $a$ to find $b$ and $c$ .

Use the given condition that $c = a + b - 7$ to express $c$ in terms of $a$ and $b$ .
$c = a + b - 7$

Substitute $c = a + b - 7$ into the perimeter equation $a + b + c = 75$ .
$a + b + (a + b - 7) = 75$

Simplify the equation to combine like terms.
$2a + 2b - 7 = 75$

Solve for $2a + 2b$ .
$2a + 2b = 82$

Divide the equation by 2 to isolate $a + b$ .
$a + b = 41$

Use the given condition $2a = c - 6$ to express $c$ in terms of $a$ .
$c = 2a + 6$

Substitute the expression for $c$ from STEP_6 into the equation $c = a + b - 7$ .
$2a + 6 = a + b - 7$

Solve for $b$ .
$2a + 6 = a + b - 7$ $2a - a + 6 + 7 = b$ $a + 13 = b$

Substitute $b = a + 13$ into the equation $a + b = 41$ from STEP_5.
$a + (a + 13) = 41$ $2a + 13 = 41$

Solve for $a$ .
$2a = 28$ $a = 14$

Use $a = 14$ to find $b$ .
$b = a + 13$ $b = 14 + 13$ $b = 27$

Use $a = 14$ to find $c$ .
$c = 2a + 6$ $c = 2(14) + 6$ $c = 28 + 6$ $c = 34$
Solution: The shortest side is $14$ cm. The middle side is $27$ cm. The longest side is $34$ cm.