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Molodets [167]
2 years ago
5

The pair of triangles that are congruent by ASA criterion is

Mathematics
1 answer:
stealth61 [152]2 years ago
4 0

Answer:

The pair of triangles that are congruent by the ASA criterion isΔ ABC and Δ XYZ.

The pair of triangles that are congruent by the SAS criterion is Δ BAC and ΔRQP.

Step-by-step explanation:

Two triangles are congruent by ASA property if any two angles and their included side are equal in both triangles .In triangles Δ ABC and Δ XYZ the equal side 5 is between the two equal angles. So these triangles are congruent by ASA criterion.

Two triangles are congruent by SAS if  two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle .In triangles  Δ BAC and ΔRQP. the included angles A and Q are equal and hence the triangles are congruent by SAS criterion.

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Step-by-step explanation:

Area of Triangle Knowing All Sides:

Sides: 50.000, 60.000, 40.000

Area  : 992.1567

Area of Triangle given by its 3 Sides:

I will show two ways to find the area. One way is very short - The 2000 years old Heron's Formula. The other method may be longer but it's more "educational" as it teaches us important analytic geometry lessons.

But before we even start, we have to verify that the Basic Triangle Inequality is satisfied.

The Basic Triangle Inequality:

The Basic Triangle Inequality states that for a triangle with side lengths a,b,c , the following is true:

    a < b + c,    b < a + c,    c < a + c

What this means, in plain English, is that for any triangle, the sum of the lengths of any two sides must be greater than the length of the remaining side.

Let us now check:

     60.000 < 40.000 + 50.000   true

     40.000 < 60.000 + 50.000   true

     50.000 < 60.000 + 40.000   true

Now that the basic inequalities are satisfied, we know that these three side lengths can make a triangle, so we can move on to calculating the area of said triangle

Heron's Formula for the area of a triangle:

The 2000 year old Heron's Formula states that the area of a triangle whose sides have lengths  a, b,  and  c  is

     SQRT (s(s-a)(s-b)(s-c))

where  s  is the semiperimeter of the triangle; that is,

     s = ( a + b + c ) / 2

Let us calculate

Area =  SQRT ( 75.000 • 15.000 • 35.000 • 25.000 ) =

           SQRT ( 984375.000 ) =

           992.1567  

Find area using the Base Height formula:

Let  B  (for base) denote the length of the longest side of the triangle

Let  h  (for height) denote the length of a perpendicular line, from the vertex opposite that side, to the side itself.

Note that  h  splits our triangle into two right-angled triangles, both having  h  as height, the base of the left triangle is denoted by  X  and the base of the right triangle is denoted by  40.000 - X

To find the area of a triangle, multiply the base by the height, and then divide by 2. In algebraic notation, Area  = 0.5 • 40.000 • h   To be able to find the Area, using this formula, we must know the value of  h

Using the Pythagorean theorem to find the Height:

Applying the Pythagorean theorem to the left right-angled triangle we get:

     h2 = (50.000)2- X2  

While the right right-angled triangle is "telling" us that:

     h2 = (60.000)2- (40.000 - X)2  

Two things which are equal to  h2 , are also equal to one another (It is a property of "Equation" that   if z=p and z=q then p=q ):

     (50.000)2- X2 = (60.000)2- (40.000 - X)2  

Expand the above and simplify :

     (40.000)2 + (50.000)2 - (60.000)2 = 80.000 • X

     500.000 = 80.000 • X

     X = 6.250

Plug this for X in:  h2 = (50.000)2- X2  

     h2 = (50.000)2- ( 6.250)2  

     h2 = (2460.938)

h = sqrt (2460.938) = 49.6078  

Put the Triangle Area Formula to use:

Finally put the formula  Area = Base * Height * 0.5   to use:

    Area  = 40.000 • 49.608 • 0.5

    Area  = 992.1567  

Note that this result is identical to the one we got using Heron's Formula !

Area of Triangle Knowing all Sides :

    Sides: 50.000, 60.000, 40.000      Area  : 992.1567

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Answer:

Jess used 40 blocks in total.

Step-by-step explanation:

First stack: 8 blocks

Second stack: First stack + 4 = 8 + 4 = 12 blocks

Third stack: Second stack + 8 = 12 + 8 = 20 blocks

Total number of blocks: First stack + Second stack + Third stack = 8 + 12 + 20 = 40 blocks

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